SPECTRAL PROPERTIES OF SCHRÖDINGER OPERATORS
DEFINED ON N -DIMENSIONAL INFINITE TREES
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
Abstract. We study the discreteness of the spectrum of Schrödinger operators which are defined on N -dimensional rooted trees of a finite or infinite
volume, and are subject to a certain mixed boundary condition. We present
a method to estimate their eigenvalues using operators on a one-dimensional
tree. These operators are called width-weighted operators, since their coefficients depend on the section width or area of the N -dimensional tree. We
show that the spectrum of the width-weighted operator tends to the spectrum of a one-dimensional limit operator as the sections width tends to zero.
Moreover, the projections to the one-dimensional tree of eigenfunctions of the
N -dimensional Laplace operator converge to the corresponding eigenfunctions
of the one-dimensional limit operator.
Contents
1. Introduction
2. Preliminaries
2.1. General notations
2.2. The tree T1
2.3. The ε-inflated N -dimensional tree
2.4. Cross sections and functions on T1 and TNε
2.5. Function spaces
2.6. Laplace and Schrödinger operators on T1 and TN
3. Behavior of functions near the vertices
4. Discreteness of the spectrum on T1 and TN
4.1. Discreteness of the spectrum for weighted operators on T1
4.2. Discreteness of the spectrum for operators on TN
4.3. Further results
5. Convergence of the spectra of width-weighted operators
6. ε-dependent bounds for the eigenvalues of N -dimensional tree
6.1. Rayleigh quotients of Schrödinger operator on T1 and TNε
6.2. T1 -based estimates for the spectrum on TNε
7. Convergence of eigenfunctions of Laplace operator on TNε
References
1
4
4
4
5
6
7
8
9
13
13
15
17
18
23
23
28
30
33
1. Introduction
Let T1 be a one-dimensional infinite tree. We assume throughout this paper
that T1 is regular (see Definition 2.1 and Remark 1.1). For N ≥ 2, we also
consider an ε-inflated tree TNε around T1 which is an N -dimensional offset (or
Date: June 6, 2006.
2000 Mathematics Subject Classification. Primary 35J10, 35P15; Secondary 34B10, 34L15.
Key words and phrases. Schrödinger operator, spectrum, quantum graph, thin domains, tree.
1
2
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
$
%
Figure 1. An example of one and two dimensional trees.
A. One-dimensional tree. B. Two-dimensional tree presented in R2 . Some of its
triangle connectors and rectangle edges are emphasized.
fattening) of T1 . See Figure 1 for an illustration of a 2-dimensional tree T2 = T21 .
We prove ε-dependent estimates for the spectrum of the eigenvalue problem
(1.1)
Lε u := −∆u + WTNε u = λε u
on H01 (TNε ),
subject to the Neumann boundary condition on ∂TNε except on the root of the
tree, where we impose the Dirichlet boundary condition. We assume that WTNε is
a bounded and continuous potential on TNε . Specifically, we show that if TNε has a
finite radius, then under some further assumptions the spectrum of Lε is discrete
and the eigenvalues of the Schrödinger operators Lε satisfy λεi → µi as ε → 0,
where µi are the eigenvalues of the following weighted Schrödinger operator on T1
1
(1.2)
Lu := − (ρu0 )0 + WT1 u = µi u.
ρ
Here ρ > 0 is a weight function on T1 defined in terms of the inflation TNε , and
WT1 is the cross section average of WTNε .
The spectral behavior of the Neumann Laplacian and Schrödinger operators
on thin domains has been extensively investigated. Indeed, in [21], Rubinstein
and Schatzman study the relation between the spectral properties of the Laplace
operator defined on a metric graph G and on a strip shaped domain Gε of width ε
around G. The results of [21] on the spectrum of the Laplacian cannot be applied
to our trees because of the following essential differences between the problems:
(1) Rubinstein and Schatzman treat the case in which the graph G has a finite
number of vertices, while our tree T1 has an infinite number of vertices.
(2) They consider graph-surrounding domains having a constant (uniform)
width. In the case of an infinite trees, the discreteness of the spectrum
imposes that the width of higher branches of the tree must be scaled.
(3) In particular, the inflated finite graph is of finite volume, while our inflated
infinite tree may have infinite volume.
In [10], Kuchment and Zeng extend the results in [21]. For example, the conditions
on the smoothness of the boundary of the domain near the vertices were relaxed
and the constant width of the surrounding domain is not assumed.
Since T1 in our case is an infinite tree, the results of [10, 21] do not apply in
our setting. Nevertheless, we were able to modify the approach in [21] to obtain
similar results in the infinite case. In particular, we could not compare directly
the eigenvalue λεi to µi . Instead, we find it more convenient to compare the
SCHRÖDINGER OPERATORS ON INFINITE TREES
3
spectra of −∆ + WTNε on TNε to the Schrödinger operator on T1 subjected to a
pair of ε−dependent weight functions ρ1,ε , ρ2,ε , satisfying ρ1,ε , ρ2,ε → ρ as ε → 0.
So, we replace (1.2) by
Lε u := −
1
(ρ1,ε u0 )0 + WT1 u = µεi u ,
ρ2,ε
and prove that the λεj is approximated, on the one hand, by µεj while the later is
approximated by µj for ε small.
Spectral properties of Schrödinger operators defined on infinite one-dimensional
metric trees and graphs has also been intensively studied. In [4], Carlson shows
that if G is a connected metric graph which has a finite total edges length (a finite
volume), then the Laplacian defined on G has a compact resolvent and therefore
a discrete spectrum. Solomyak and Naimark have developed general tools for
studying spectral properties of Schrödinger operators on metric graphs and trees
(see, for example, [14, 15, 22, 23]). In [22], Solomyak has proved that if T1 is
a regular tree whose radius is finite, and if WT1 (x) is a radial measurable real
valued function which is bounded below, then the spectrum of L is discrete.
Solomyak’s result is stated for trees of uniform weight function ρ and its proof
relies on the monotonicity of g, where g(t) is the number of branches which
contain points of distance t from the root. In fact, to adjust Solomyak’s proof
for our case, one needs to assume only that gρ is a monotone nondecreasing. If
ρ is constant then it is a natural assumption, but if ρ(t) is decreasing (as in our
case), this monotonicity may be violated. So, we extend this result under a milder
condition on gρ.
We prove the discreteness of the spectrum of Schrödinger operators on regular
N -dimensional trees with infinite volume, as long as the tree radius is finite. Our
proof relies on a lemma of Lewis [11, Lemma 1]. The proof of the discreteness in
the N -dimensional case can be applied also to show that the L2 -norm of functions
which are bounded in H01 (TNε ) does not accumulate at the tree connectors or ends.
A natural question emerging from the correspondence between the eigenvalues
of N -dimensional Laplace operator, and one-dimensional width-weighted operators, is whether the corresponding eigenfunctions present the same convergence
behavior. In [7, 8], Kosugi has proved that solutions of (semilinear) elliptic equations on finite N -dimensional trees indeed converge as the width tends to zero to
solutions of width-weighted equations. We present a different method and prove
that certain projections of eigenfunctions of the Laplace operator on TNε converge
to the corresponding eigenfunctions on T1 . In contrast to [7, 8], we treat infinite
trees rather than trees with a finite number of vertices. In addition, our assumptions on the smoothness of the connectors are much weaker than those in [7, 8],
and in fact, we require only that the connectors have a Lipschitz boundary.
Remark 1.1. Our method applies to more general setting. But to facilitate the
presentation, we restrict our study in the present paper to the case where T1 is
a regular metric tree, and the inflated N -dimensional tree is a self-similar radial
tree with ‘cylindrical’ edges.
We wish to mention two more articles which study the spectrum of thin domains. In an earlier article [9], Kuchment and Zeng study the dependence of
the spectrum of the Neumann Laplacian on the behavior of the surrounding thin
domain near the vertices. They found differential operators on the graph which
correspond to the case in which the neighborhoods of the vertices are much larger
4
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
or much smaller than the tubes connecting them. In [5], Evans and Saito proved
results about the connection between the essential spectrum of the Neumann
Laplacian on thin domains surrounding trees and the essential spectrum of their
skeletons. They apply their results on horns, spirals, “rooms and passages” domains and domains with fractal boundaries. In our case the essential spectrum
is empty, as was mentioned above.
The motivation for our problem is that fractal structures, and in particular,
fractal tree-like structures, have a vast applications range. For example, fractal
geometry is used in order to form antennas, which present a multi-band behavior
(see [1, 18]). In [19], Puente et al. state that fractal tree shaped antennas have
a denser band distribution than previously reported Sierpinski fractal antennas.
Estimating the eigenvalues of the Laplace operator defined on such domains may
help in specifying the natural transmission frequencies for the antennas.
Another applications field for fractal geometry is medical modelling. Nelson et
al. mention in [16] that fractal models can be applied to human lungs, vascular
tree, neural networks, urinary ducts, brain folds and cardiac conduction fibers.
Fractal models of human lungs can be found also in [12, 17, 24].
The outline of this article is as follows. In Section 2, we present the basic
notations we use, describe the class of trees we are interested in, and define the
operators on the trees. Section 3 is devoted to the study of the behavior of
H 1 -functions near the vertices. In Section 4, we prove the discreteness of the
spectrum of Schrödinger operators on T1 and TN . The convergence (as ε → 0) of
the spectrum of {Lε }, the operator sequence defined on T1 , to the spectrum of
the limit operator L is proved in Section 5.
In sections 6.1.1 and 6.1.2 we define transformations between H01 (TN ) and
1
H0,ρ
(T1 ) and prove comparison theorems for the Rayleigh quotients of the one
2
and N -dimensional operators. In Section 6.2, we use these comparison theorems
to characterize the behavior of the spectrum on TN . Finally, the convergence of
projections of N -dimensional eigenfunctions of Laplace operator to eigenfunctions
of the one-dimensional width-weighted operators is proved in Section 7.
2. Preliminaries
2.1. General notations.
(1) Throughout the article, c, c1 , c2 , . . ., and C denote constants, whose exact
values are irrelevant, and may change from line to line.
(2) Let {aj } and {bj } be positive sequences. We denote aj ³ bj if there exists
a constant c > 0 such that c−1 ≤ aj /bj ≤ c for all j ∈ N. We use a
similar notation for positive functions, i.e., we denote f ³ g if there exists
a constant c > 0 such that c−1 ≤ f (x)/g(x) ≤ c for all x in the domain of
the functions f and g.
(3) For a domain Ω ⊂ RN , we denote by |Ω| its volume in RN .
2.2. The tree T1 .
(1) T1 is a one-dimensional connected rooted metric tree. It contains an infinite number of vertices v, connected by edges e.
(2) The root O of T1 is a distinguished (and unique) vertex. Its generation
number is defined to be zero.
(3) A vertex of T1 is of generation j if it is connected to the root by a succession
of j edges. The generation of a given vertex v is denoted by gen(v).
SCHRÖDINGER OPERATORS ON INFINITE TREES
5
(4) Likewise, e is an edge of generation j if it connect a pair of vertices of
generations j and j + 1, respectively. The generation number of a given
edge e is denoted by gen(e).
(5) The Euclidian length of an edge e is denoted by |e|.
(6) The degree of a vertex v is k(v). It is the number of edges connecting v
to the vertices of generation gen(v) + 1.
(7) The set of all edges meeting at a vertex v is N (v). There are exactly
k(v) + 1 edges in N (v).
(8) The distance dist(x, y) between x, y ∈ T1 is the Euclidian length of the
path on T1 connecting x to y. We denote |x| := dist{O, x}.
(9) g(t) is the counting function of T1 , namely, g(t) is the number of edges
which contain a point x ∈ T1 with |x| = t.
P
(10) R(T1 ) ≡ supx∈T1 |x| is the radius of T1 . L(T1 ) ≡ e∈T1 |e| is the length of
T1 .
Definition 2.1. T1 is called radial if the length |e| of each edge e and the degree
k(v) of each vertex v depend only on gen(e) and gen(v), respectively. A radial
tree is called regular if k(v) = k is a constant, independent of the generation.
2.3. The ε-inflated N -dimensional tree. The tree T1 defined above is, in fact,
a combinatorial object, but we always treat it as a metric tree or quantum graph.
We shall now describe a way to construct an N -dimensional manifold which is an
ε-inflation of T1 . For simplicity we shall assume that T1 is radial and regular.
(1) A Lipschitz domain Ω ⊂ RN −1 is given. It corresponds to the (scaled)
cross section of the edges. We take the origin of RN −1 to be an interior
point of Ω, called the center of Ω.
(2) A Lipschitz domain V ⊂ RN is given. It corresponds to the (inflated)
vertices. We take the origin of RN to be an interior point of V , called the
center of V .
(3) 0 < δ < 1 is the scaling factor. The notation δΩ stands for the scaled
domain δΩ := {δx | x ∈ Ω}. Similarly δV := {δx | x ∈ V }.
(4) The boundary of V contains k+1 disjoint sections: One of these sections is
an isometric image of Ω, denoted by S0 . The other k sections are isometric
images of δΩ, and denoted by S1 , . . . Sk .
(5) The orthogonal projections of the center of V into S0 and Sj ⊂ ∂V for
1 ≤ j ≤ k coincide with the isometric image of the centers of Ω and δΩ,
respectively.
Next, we define the inflated tree TNε . For this, let us consider a certain embedding
of T1 in RN +1 . We denote this embedding of T1 by the same name, T1 . It is, in
fact, determined by the choice of the inflated vertex V , to be explained below:
(6) For each vertex v in the embedded tree T1 , the inflated vertex is an isometric image of V ε (v) := εδ gen(v) V whose center coincides with v.
(7) Each edge e ∈ N (v) is perpendicular to Seε (v), where Seε (v) is the isometric
image of the section of ∂V ε (v) intersecting the edge e.
ε
(8) The skeleton of V ε (v) is V (v) := V ε (v) ∩ T1 .
(9) For each edge e of the embedded T1 , the inflated edge is
E ε (e) = e × Seε (v) \ ∪v V ε (v) .
ε
(10) The skeleton of E ε (e) is E (e) := E ε (e) ∩ T1 .
An inflated 2-dimensional tree is depicted in Figure 2. A somewhat degenerate
6
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
C A ε (B
@
ε
)
9 ε (Y)
(? )
> ε< = ;
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H
H Iε G
(
)
:
ε
H Fε G
(
)
Ωε
(9 )
E ε (D )
M -L , =M
6 P5 O =6
- L , N K J P5 O R Q
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8 ε
6 (5 )
` _
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p e = S] v ^ e
!! 7 ε
ε
1
ε
ε
ε
1
ε
1
ε
ε
ε
ε
ε
Figure 2. Notations of parts of T1 and Tnε .
example of an inflated tree is the straightened tree, which we denote by T̂N . We
use T̂N as a canonical representation for TN in Section 4.2.
Definition 2.2 (The straightened tree). The inflated vertex V̂ is given by the
cylinder Ω̂ × [0, −1]. The section Ŝ0 := Ω̂ × {0} is the top of V̂ , and its base
Ω̂ × {−1} consists of k disjoint isometric copies of (k)−1/N Ω̂ × {−1}, corresponding to the sections Ŝ1 , . . . Ŝk . A two-dimensional straightened tree is depicted
in Figure 3. The above condition implies that Ω̂ is a box in RN of a certain type which depend on k and N . Indeed, take a box Ω̂ whose sizes are
(1, k 1/N, k 2/N, . . . , k (N −1)/N ), then k copies of (k)−1/N Ω̂ exactly cover Ω̂. We may
of course consider also other tilings.
Corollary 2.3. The straightened tree T̂N can be parameterized by the cylinder
Ω̂ × R̂, where R̂ is its radius (see Figure 3).
2.4. Cross sections and functions on T1 and TNε . There is a natural coordinate system on each of the edges E ε (e) ⊂ TNε , namely ~x ∈ E ε (e) ⊂ TNε is
parameterized as ~x = x = (~s, θ), where ~s is a parameterization of the corresponding perpendicular section Se in Ω, scaled by εδ gen(e) , and θ is a parameterization of
ε
e\∪v V (v). We can also use the natural parameterization of V , scaled by εδ gen(v) ,
to describe the coordinate system in the inflated vertex V ε (v). We always take
the center of V ε (v) as the origin 0 ∈ V .
SCHRÖDINGER OPERATORS ON INFINITE TREES
ba
7
c ˆ (0)
0
d ˆ (0)
ea
Figure 3. The straightened tree, T̂3 for k = 3, N = 3.
(1) We denote by fe the restriction of a function f on T1 to an edge e. In
most cases we omit this notation and write simply f instead of fe .
(2) The function ρ∗ is defined on T1 by ρ∗e = δ (N −1)gen(e) |Ω|.
(3) Let f be a function on TN1 . We denote by f ε the following rescaling of f
on TNε :
(
f (θ, ~s/ε) ~x = (θ, ~s) ∈ E ε (e),
ε
f (~x) = fTNε (~x) :=
f (~x/ε)
~x ∈ V ε (v),
(4) The total cross section of TN is defined for t ∈ T1 ⊂ TN as H(t) =
g(t)ρ∗ (t), where g is the counting function of the skeleton T1 of TN and
ρ∗ as defined in (2) above.
2.5. Function spaces.
(1) Let ρ > 0 be a measurable (weight) function on T1 . Denote
½
¾
Z
2
L2,ρ (T1 ) = f | f is measurable on T1 and
|f | ρ dθ < ∞ .
T1
R
The space L2,ρ (T1 ) equipped with the inner-product < f, g >ρ := T1f gρ dθ
is a Hilbert space.
(2) C 1 (T1 ) is the space of continuous functions f on T1 , such that fe ∈ C 1 (e)
for each edge e. Let ρ > 0 be a measurable function on T1 . Hρ1 (T1 ) is the
completion of the space
)
(
XZ ¡
¢
|(fe )0 |2 + |fe |2 ρe dθ < ∞
f ∈ C 1 (T1 ) |
e∈T1
e
¤1/2
£P
R
0 2
2
.
with respect to the norm ||f ||Hρ1 (T1 ) :=
e∈T1 e (|fe | + |fe | ) ρe dθ
1
1
1
(3) H0,ρ (T1 ) is the completion in Hρ (T1 ) of C0 (T1 ). For the weight function
1
1
ρ∗ , we abbreviate H0,∗
(T1 ) := H0,ρ
∗ (T1 ).
8
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
(4) H 1 (TN ) is the completion of the space
½
¾
Z
¡
¢
1
2
2
f ∈ C (TN ) |
|∇f | + |f | dx < ∞
TN
(5)
with respect to the norm ||f ||H 1 (TN ) :=
H01 (TN )
f |O×Ω0
1
hR
2
TN
2
(|∇f | + |f | ) dx
1
i1/2
.
is the completion in H (TN ) of all functions in C (TN ) satisfying
= 0.
2.6. Laplace and Schrödinger operators on T1 and TN . We define a family
of operators on T1 using the standard definition of operators on T1 (see [21, 22]).
Let W ∈ L∞ (T1 ) be a bounded real valued potential, and let ρα and ρβ be
positive bounded L1loc (T1 ) weight functions, which satisfy ρα ³ ρβ . In particular,
1
1
1
H0,ρ
(T1 ) and H0,ρ
(T1 ) are equivalent in the sense that u ∈ H0,ρ
(T1 ) if and only
α
α
β
1
if u ∈ H0,ρβ (T1 ), and there exists a constant c > 0 independent of u such that
1
1
1
1
||u||H0,ρ
(T ) .
(T1 ) ≤ ||u||H0,ρ
(T1 ) ≤ c||u||H0,ρ
α 1
α
β
c
We denote by
¸
X Z · ρα
0
0
(ue ) (v e ) + W ue v e ρβ dt
E(u, v) :=
ρ
β
e
e∈T
1
1
1
the bilinear form on H0,ρ
(T1 ) × H0,ρ
(T1 ). Without loss of generality, we may
β
β
1
assume that E ≥ 0 on C0 (T1 ), so, E is a symmetric and nonnegative closed
1
bilinear form, and H0,ρ
(T1 ) is dense in L2ρβ (T1 ). By Friedrich’s extension theoβ
rem (see e.g. Theorem X.23 in [20]) or the First Representation Theorem (see
Theorem VI.2.1 in [6]), there exists a unique selfadjoint operator Lα,β such that
Dom(Lα,β ) ⊆ Dom(E) and E(u, v) =< Lα,β u, v >ρβ for all u ∈ Dom(Lα,β ) and
1
(T1 ). By this theorem, the domain of Lα,β is given by:
v ∈ H0,ρ
β
1
Dom(Lα,β ) = {u ∈ H0,ρ
(T1 ) | |E(u, v)| ≤ C|v|L2ρ
β
β
(T1 )
1
∀v ∈ H0,ρ
(T1 )}
β
for some constant C. Moreover, it is well known (see e.g. [21]) that the domain of
Lα,β is contained in the space of all functions u satisfying the following Kirchhoff
conditions:
1
⊂ C(T1 ) ).
(1) u is continuous at the vertices (since H0,ρ
β
P
0
(2) e∈N (v) (ρα )e (ue (v)) = 0 in each vertex v ∈ T1 .
We will call operators of this form width-weighted operators, because we will
use them for weights ρα and ρβ which are closely related to the width or section
area of TN . Similar operators are also presented by Evans and Saito in [5].
1
Remark 2.4. The domain of the operator Lα,β is clearly dense in H0,ρ
for ρ = ρα
or ρ = ρβ .
Finally, the Laplace operator on the tree TN is defined by the Friedreich extension of the quadratic form
Z
(2.1)
EN (u, w) :=
∇u · ∇w̄ dx
TN
for u, w in the space H01 (TN ) (see the definition of H01 (TN ) in Section 2.5 §(5)).
SCHRÖDINGER OPERATORS ON INFINITE TREES
9
3. Behavior of functions near the vertices
Here we concentrate on a neighborhood of a vertex (resp. an inflated vertex)
ε
in T1 (resp. TNε ). For T1 , we shall consider the skeleton V (v) corresponding to
a vertex v, as defined in Section 2.3 §(8). We shall also denote the “canonical”
skeleton, corresponding to ε = 1, by V (v). Occasionally, we shall omit the
reference to a particular vertex v and just denote it as V . The end points of V (v)
are denoted by pe , where e ∈ N (v) (see Figure 2). Recall that ρ∗ , as defined in
Section 2.4 §(2), is a positive weight function on T1 , which is constant on each
edge.
(1) For each edge e ∈ N (v) define a nonnegative function ψ(e) ∈ C 1 (V ) such
that ψ(e) (pe ) = 1 and ψ(e) (pẽ ) = 0 for ẽ 6= e. We also assume that
X
(3.1)
ψ(e) = 1 on V (v) .
e∈N (v)
©
ª
If the skeleton V is scaled by δ > 0, so V → δV := δθ | θ ∈ V , where the
δ
vertex v is taken as the origin, then ψ(e) is scaled into ψ(e)
(x) := ψ(e) (x/δ)
for any x ∈ δV .
(2) Let V be the “canonical” inflated vertex defined in Section 2.3 §(2). We
choose a family of nonnegative functions φ(e) ∈ C 1 (V ) ∩ C(V̄ ) such that
(
1 θ(x) ∈ Se ,
φ(e) (x) =
0 θ(x) ∈ Sẽ , where ẽ 6= e,
and
X
(3.2)
φ(e) = 1
on V.
e∈N (v)
Similarly, if V is scaled by δ > 0, so V → δV := {δx | x ∈ V }, where
the center of V is taken as the origin, then φ(e) is scaled into φδ(e) (x) :=
φ(e) (x/δ) for any x ∈ δV .
(3) Next, define for each e ∈ N (v) the quadratic (k + 1) × (k + 1) matrices:
Z
Z
0
0 ∗
Al,m := (ψ(l) ) (ψ(m) ) ρ dθ, Al,m :=
∇φ(l) · ∇φ(m) dx,
V
and
Bl,m :=
V
Z
∗
ψ(l) ψ(m) ρ dθ,
V
Bl,m :=
Z
φ(l) φ(m) dx.
V
¡ √
√
¢
(4) Let ~1 := 1/ k + 1, . . . 1/ k + 1 ∈ Rk+1 , and for any f~ ∈ Ck+1 denote
³
´
(3.3)
f~x~1 := f~ − f~ · ~1 ~1,
where · is the standard inner product in Ck+1 .
The following Lemma is elementary, but essential for our analysis.
Lemma 3.1. The matrices A and A are nonnegative definite, and B and B are
strictly positive definite. In particular, there exist constants α A > 0, αA > 0,
αB > 0, and αB > 0, such that
1 ~~2
1 ~~2
(3.4)
|f x1| ≤ f~ · Af~∗ ≤ αA |f~x~1|2 ,
|f x1| ≤ f~ · Af~∗ ≤ αA |f~x~1|2 ,
A
αA
α
10
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
and
1 ~2
1 ~2
~ · Bf~∗ ≤ αB |f~|2 ,
|f | ≤ f~ · Bf~∗ ≤ αB |f~|2
|
f
|
≤
f
B
B
α
α
for all f~ ∈ Ck+1 , where f~∗ denotes the complex conjugate of f~t .
(3.5)
Proof. The non-negativity (resp. positivity) of A and A (resp. B and B) follows
from the corresponding definitions, while (3.4) follows from (3.1) and (3.2). ¤
Let us introduce the following functionals on H 1 (V ):
Z
V
(3.6)
I γ [g] := (|g 0 |2 + γ|g|2 )ρ∗ dθ
for γ = 0, 1,
V
and for f~ ∈ Ck+1 let us denote:
AV ,f~ = {g ∈ H 1 (V ) | g(pe ) = fe , e ∈ N (v) }.
(3.7)
Lemma 3.2. Using the notations (3.6) and (3.7), we have for γ = 0, 1 that
V
V
J γ [f~] := inf I γ [g]
g∈AV ,f~
is attained by a unique function h, which solves the Dirichlet problem
−h00 + γh = 0
(3.8)
in V ∩ e, h(pe ) = fe
∀e ∈ N (v),
and satisfies Kirchhoff ’s conditions
X
(3.9)
ρ∗e h0e (v) = 0.
e∈N (v)
V
V
Proof. The existence of minimizers u for I 0 and I 1 , which satisfy (3.8) is standard
(see e.g. the proof in [5, Theorem 2, pp. 448–449]).
V
We need to prove that the minimizer u of I γ satisfies Kirchhoff’s derivatives
condition. To this end, let v ∈ C01 (V ) and 0 6= ² ∈ R. Since u is a minimizer,
Iγ [u] ≤ Iγ [u + ²w], and therefore,
Z
(u0 w 0 + γuw)ρ∗ dθ = 0.
V
2
By elliptic regularity u ∈ C (V ∩ e); Moreover, u is continuous in V . Recall that
00
ρ∗ is constant on each edge, therefore, −u + γu = 0 on V ∩ e. Thus,
X Z
(3.10) 0 =
(u0 w 0 + γuw)ρ∗ dθ
e∈N (v)
V ∩e
¯ve
¯
X
X Z
¯
∗
0
¯
=
ρe (ue ) we ¯ +
(−u00 + γu)wρ∗ dθ = w(v)
ρ∗e u0e (v).
¯
e∈N (v)
e∈N (v)
e∈N (v) V ∩e
X
pe
V
V
The uniqueness of the minimizers of I 0 and I 1 follows since both are minima of
strictly convex functionals on the underlying domains.
¤
Lemma 3.3. There exist β A > 0 and β B > 0 such that for all δ > 0
(3.11)
δV
I 0 [f~] ≥ δ −1 β A |f~x~1|2 ,
SCHRÖDINGER OPERATORS ON INFINITE TREES
11
and
δV
I 1 [f~] ≥ δ −1 β B (|f~x~1|2 + δ 2 |f~|2 ).
(3.12)
Proof. In the following, we use the notations introduced in Lemma 3.2, and in
(3.6) and (3.7). Consider the case δ = 1 first. Let {~σe } be the standard basis
vectors in Ck+1 , where e ∈ N (v). Let h(e) ∈ H 1 (V ) be the unique minimizer of
V
J γ [σe ]. By Lemma 3.2 it follows that


Z
X
X
¤
£
V
V
~


fe fẽ
h(e) h0(ẽ) + γh(e) h(ẽ) ρ∗ dθ,
fe h(e) =
J γ [f ] = I γ
e,ẽ∈N (v)
e∈N (v)
V
where each h(e) satisfies
−h00(e) + γh(e) = 0
in V ,
h(e) (pe ) = 1, h(e) (pẽ ) = 0 ∀ẽ 6= e.
Let γ = 0. By Lemma 3.2, J 0 [f~] is attained uniquely by the harmonic function h
which solves the corresponding Dirichlet problem (and satisfies Kirchhoff’s conditions). In particular, it depends only on f~ and the domain VP
. Since each solution
h satisfying h(pe ) = fe can be presented uniquely by h = e∈N (v) fe h(e) , it follows that J 0 [f ] is a bilinear form. Clearly, it is a nonnegative k + 1 dimensional
form whose kernel contains only constant multiplicities of ~1 for which the unique
solution of the Dirichlet problem is constant. Therefore, it is equivalent to all
nonnegative forms with such a kernel, and in particular, to |f~x~1|2 .
The proof for the case γ = 1 is similar except for replacing the Laplace operator
by the operator −d2 /dθ2 + 1 and |f~x~1|2 by |f~|2 .
Now, if δ < 1 and γ = 0 we observe that the harmonic minimizers h(e) are
scaled into h(e) (·/δ), and
Z
Z
0
0
∗
−1
h(e) (x/δ)h(ẽ) (x/δ)ρ dθ = δ
h0(e) h0(ẽ) ρ∗ dθ.
δV
V
For δ < 1 and γ = 1, we use similar scaling argument to obtain (3.12).
¤
We wish to prove now the analog of Lemma 3.3 for the N -dimensional case.
Consider the following functionals for γ = 0, 1 :
Z
(3.13)
Iγ [g] := (|∇g|2 + γ|g|2 ) dx.
V
1
For all h ∈ H (V ) and 0 ≤ j ≤ k we denote the average of h on the section
Sj ⊂ ∂V by
Z
1
(3.14)
Pj (h) :=
h ds
|Sj | Sj
(see Section 2.3 §(4)). For F~ ∈ Ck+1 we define
©
(3.15)
AV,F~ := g ∈ H 1 (V ) | Pj (g) = Fj
ª
∀j = 0, ..., k .
Lemma 3.4. Let F~ ∈ Ck+1 . Using the above notations, we have for γ = 0, 1 that
Jγ [F~ ] := inf Iγ [g]
g∈AV,F~
is attained by a function h, which is the unique solution of the problem
(3.16)
−∆h + γh = 0 in V,
h ∈ AV,F~ ,
12
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
and satisfies weakly the mixed boundary conditions
∂h
∂h
(3.17)
= 0 on ∂V \ ∪kj=0 Sj ,
and
= κj on Sj ,
∂n
∂n
where κj for j = 0, ..., k are uniquely determined constants.
Proof. The proof of (3.16) for the case γ = 1 is standard. Indeed, let {w i }∞
1=1 be
∞
~
a minimizing sequence satisfying limi→∞ I1 [wi ] = J1 [F ]. Then {wi }i=1 is bounded
in H 1 (V ). Therefore, there exists a subsequence {wi } and a function v ∈ H1 (V )
such that wi * v in H 1 (V ).
Since Pj (f ) is a continuous functional on H 1 (V ) in the strong topology, it is
also continuous in the weak topology. In particular, AV,F~ is closed in the weak
topology of H1 (V ) so v ∈ AV,F~ . The lower semicontinuity of I1 implies that
I1 [v] = J1 (F~ ). Moreover, v is unique because I1 is convex.
It remains to prove that v satisfies the boundary conditions in (3.17). Since v
is a minimizer in AV,F~ , it follows that
Z
Z
Z
∂v
∂v
0 = (∇v · ∇w + vw) dx =
w
dξ +
w
ds.
∂n
∂n
V
∂V \∪kj=0 Sj
∪kj=0 Sj
− . The first term in the last expression is thus zero only if
for any w ∈ AV,→
0
∂v/∂n = 0 on ∂V \∪kj=0 Sj in the weak sense. Since the average of the test function
w is zero on each sector (Pj (w) = 0), the second term is zero if ∂v/∂n = κj (in
the weak sense) on Sj . Finally, the multipliers κj are uniquely determined due
to the uniqueness of v for any F~ .
The proof of (3.17) for the case γ = 0 is similar, except that we have to prove
the bound in L2 (V ) of the minimizing sequence. Since V is a bounded Lipschitz domain, by [13, Theorem 5.5.1, and the remark in p. 286], the embedding
H 1 (V ) → L2 (V ) is compact. Hence, the spectrum of Helmholtz operator with the
Neumann boundary condition for such domains is discrete. Its first eigenvalue is
1, and is a simple eigenvalue corresponding to the constant ground state. Hence,
the Poincaré inequality
Z
Z
2
−1
|∇v|2 dx.
(3.18)
|v| dx ≤ Λ2
V
V
holds for all functions v perpendicular to the constant in H1 (V ), where Λ2 is the
second eigenvalue of the Neumann Laplacian on V .
We now repeat the argument for the case γ = 1, but restrict our domain to
the domain of all functions in v ∈ AV,F~ which are perpendicular to the constant.
The minimizer u obtained in this way satisfies Pj (u) = Fj + κ for some κ ∈ R
and j ∈ {0, . . . , k}. Then u − κ ∈ AV,F~ .
¤
Let now δ > 0, and set δV := {δx ; x ∈ V } the scaled inflated vertex, where
we assume (as usual) that the center of V is in the origin. The sections of δV
are scaled accordingly, and we denote them by δSj , 0 ≤ j ≤ k. We define,
correspondingly, the averaging operator on δSj for h ∈ H 1 (δV ):
Z
1
δ
(3.19)
Pj (h) := N −1
h ds,
δ
|Sj | δSj
and
(3.20)
©
AδV,F~ := g ∈ H 1 (δV ) | Pjδ (g) = Fj
∀j = 0, ..., k
ª
.
SCHRÖDINGER OPERATORS ON INFINITE TREES
13
Using Lemma 3.4, the following lemma is proved analogously to the proof of the
second part of Lemma 3.3.
Lemma 3.5. There exist β A > 0 and β B > 0 such that for all f ∈ AδV,F~
Z
(3.21)
|∇f |2 dx ≥ δ N −2 β A |F~ x~1|2 ,
δV
and
(3.22)
Z
2
δV
2
(|∇f | + |f | ) dx ≥ β
B
³
δ
N −2
´
2
N ~ 2
~
~
|F x1| + δ |F | .
4. Discreteness of the spectrum on T1 and TN
In this section we study the discreteness of the spectrum of width-weighted
operators on T1 and Schrödinger operators on TN .
4.1. Discreteness of the spectrum for weighted operators on T1 . In [4],
Carslon has shown that the spectrum of the Laplacian on a connected metric
graph G of finite volume which has a compact completion G is purely discrete.
Solomyak [22] has extended Carlson’s result to regular trees of a finite radius R.
Theorem 4.1 (Solomyak [22]). Let T1 be a radial tree such that R(T1 ) < ∞ and
its branching function is uniformly bounded. Let W (x) be a radially symmetric
measurable real valued function which is bounded below. Then the spectrum of
−∆ + W on T1 is purely discrete.
Outline of Solomyak’s proof. Solomyak constructed a family of weighted operators {AW,v } which are defined on the intervals [tv , R) ⊆ R, where tv is the distance
of a vertex v from the root O. The operators AW,v are defined as the selfadjoint
operators in L2g (tv , R), associated with the quadratic form
Z R
£ 0 2
¤
|u (t)| + W (t)|u(t)|2 g(t) dt
u ∈ C0∞ (tv , R),
(4.1)
aW,v [u] :=
tv
where g is the counting function. Using a decomposition of functions in H(T1 ) into
symmetric functions on subtrees [14] (which implies the spectral decomposition
of the Laplacian to these operators), Solomyak showed the equivalence between
the discreteness of the spectrum of the Laplacian on T1 and the discreteness
of the spectrum of AW,v on [tv , R) for all vertices v ∈ T1 . Using a theorem of
Birman and Borzov [3] and a certain change of variables, it is then shown that
all the operators AW,v have discrete spectra. The proof of this part relies on the
monotonicity of the counting function g.
¤
The basic ingredient in Solomyak’s proof, namely the spectral decomposition
into the subspaces of functions which are symmetric on subtrees, still holds if
one adds weight functions which are symmetric in generations (see [14, 22] for
details). The Schrödinger-type operators we consider in this section are defined
on the weighted tree T1 and involve a pair of symmetric weight functions ρα , ρβ
and a symmetric potential W :
µ
¶
du
−1 d
(4.2)
Lα,β u := −ρβ
ρα
+ W u.
dt
dt
The spectral decomposition of Lα,β is obtained by reducing these operators to the
space of functions which are symmetric on all subtrees. The restriction of L α,β
14
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
to the symmetric subtree T1,v with a root v ∈ T1 are obtained by the quadratic
form
Z R
£ 0 2
¤
(4.3)
aα,β,W,v [u] =
|u (t)| ρα + W (t)|u(t)|2 ρβ g(t) dt.
tv
and the associated operator in L2 ([tv , R)) is denoted by Aα,β,W,v . To extend the
result of Solomyak to the weighted tree we should show that Aα,β,W,v has a discrete
spectrum for each vertex v ∈ T1 . Even though (4.3) seems very close to (4.1),
the counting function g in (4.1) is replaced by gρα and gρβ in (4.3), and these
functions are not necessarily monotone. We prove the discreteness of L α,β under
the weaker condition that gρα and gρβ are uniformly bounded from below:
Theorem 4.2. Let T1 be a one-dimensional tree, whose radius R is finite. Assume that 0 < ρ < 1 is a symmetric weight function on T1 , that ρα ³ ρ and
ρβ ³ ρ are symmetric weight functions. Suppose that there exists a constant
C > 0 so that
(4.4)
Cg(s)ρ(s) < g(t)ρ(t)
for all
s ≤ t ≤ R(T1 ).
Then the spectrum of the width-weighted operator Lα,β on T1 is purely discrete.
We use the following general Lemma of Lewis.
Lemma 4.3 ([11, Lemma 1]). Let D be a domain in RN . Let h be a strictly
positive symmetric closed form whose domain Hh (D) is dense in the Hilbert space
L2w (D) for a positive weight function w on D.
Suppose that D is the union of an increasing sequence of open sets {Dj }, for
which the identity injection ij : Hh (Dj ) → L2w (Dj ) is compact.
If there is a positive-valued function p(x) on D and a sequence of positive numbers
εj → 0 as j → ∞ such that
w(x)p(x)−1 < εj
and
(4.5)
Z
D\Dj
for almost every x ∈ D\Dj ,
p(x)|u(x)|2 dx ≤ h[u, u]
for all u ∈ Hh ,
then the selfadjoint operator on L2w (D) associated with the Friedrich extension of
h has a purely discrete spectrum.
Proof of Theorem 4.2. We only need to show that for any v ∈ T1 the operator
Aα,β,W,v associated with (4.3) has a discrete spectrum. Evidently, it is enough to
show it for v = O. For this, we use Lemma 4.3 with the quadratic form h = aα,β
on L2ρβ = L2 ((0, R), ρβ dt). We set D = [0, R), Dj = [0, tj ). We denote
p(θ) :=
ρ(θ)g(θ)
.
R(R − |θ|)
Since 0 < ρ < 1 and gρ satisfies (4.4), it follows that p satisfies the assumptions
of Lemma 4.3. By our assumptions ρα ³ ρβ ³ ρ, therefore it is sufficient to prove
that for all u ∈ C01 ([0, R)) and 0 < j < R we have
Z R
Z R
2
p(θ)|u(θ)| dθ ≤ C
|u0 (θ)|2 ρ(θ)g(θ) dθ.
j
0
SCHRÖDINGER OPERATORS ON INFINITE TREES
In fact, for any j < θ < R
¯Z
¯
|u(θ)| = ¯¯
R
2
Then
Z R
j
2
p(θ)|u(θ)| dθ ≤
1
≤
R
Z
R
j
·
Z
θ
|R − θ|p(θ)
ρ(θ)g(θ)
R
0
R
j
¯2
Z
¯
u (t)dt¯¯ ≤ |R − θ|
Z
θ
R
·Z
θ
¸
|u0 (t)|2 dt.
¸
2
|u (ζ)| dζ dθ
C
|u | dζ dθ ≤
R
0 2
θ
R
0
15
Z
R
j
·Z
¸
R
0 2
θ
ρg|u | dζ dθ
≤C
Z
R
0
ρg|u0 |2 dθ .
¤
4.2. Discreteness of the spectrum for operators on TN . As we have mentioned, we are interested in spectral properties of Schrödinger operators on the
N -dimensional tree TNε . It is well known that the Laplacian on a compact manifold with a smooth boundary, and with standard (regular) boundary conditions
has a pure point spectrum. However, since we wish to address also the problem of
the discreteness of the spectrum for nonsmooth trees with an infinite volume, we
cannot implement the classical theory. Instead, we use Lemma 4.3 to prove the
discreteness of the spectrum of Schrödinger operator on TNε with a finite radius.
Recall Definition 2.2 of the straightened tree T̂N . By Corollary 2.3 we can
assign T̂N a global coordinate system to the tree, namely (~s, θ), where ~s ∈ Ω̂ and
θ ∈ [0, R̂). We pose the following assumption.
Assumptions 4.4. There exists a C 1 -diffeomorphism G : TN → T̂N . We denote
by F its inverse, so that, F : T̂N → TN . Denote by J the Jacobian of F . We
assume that there is a constant C > 0 such that
¯
¯
¯ ∂F (~s, θ) ¯
¯
¯
(4.6)
∀(~s, θ) ∈ T̂N ,
¯ ∂θ ¯ ≤ C
(4.7)
0 < J(~s, θ1 ) ≤ CJ(~s, θ2 )
∀θ1 ≤ θ2 .
We have in mind the following two-dimensional example.
Example 4.5. Let T2 be a two-dimensional binary symmetric tree constructed
by gluing rectangles and triangles (see Figure 4). Let the length of a rectangle
in generation j be r j and its width be dj , where r, d ∈ (0, 1). Clearly, T2 can
be embedded in R3 to avoid overlapping of the edges. Notice that such a tree
may have an infinite
its radius is finite). Indeed, the area of such
P∞area (though
j
a tree is given by j=1 [(2dr) + βd2j ] for some constant β, hence for any choice
of r < 1, d < 1 such that rd > 1/2, the area is infinite. Let us denote by Pj the
pentagon constructed by gluing a rectangle and triangle in the j generation, and
by Pj,l for l = 1, 2 its partition into two symmetric quadrangles. We assume that
the coordinates of the vertices of the quadrangle Pj,l , (x1,a , x2,a ) for a = 1, ..., 4,
are given (up to translations) by (0, 0), (dj , 0), (dj , r j ), (0, r j + cdj ) respectively for
a constant c.
LetP
p = (R − 1)/R, where R is the radius of the original tree (p is chosen such
j
that ∞
j=0 p = R). In particular, r < p < 1. A transformation of a rectangle
16
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
l k ,0) f
l k mk
(0, r n + oqp n )
f
(1 / 2 ,0)
1
( , )
2
g
(0,0)
(0,0) (
(0,
jg
)
hih
(1/ 2 ,
)
s
θ
Figure 4. The transformation of T2 to Tb2 .
whose vertices are at (0, 0), (1/2j , 0), (1/2j , pj ), (0, pj ) onto Pj,l can be written in
the form
rj
dj
2j dj
x1 (θ, s) = (2d)j s, and x2 (θ, s) = j θ + c j θ − c j sθ.
p
p
p
An elementary calculation shows that if d ≤ p, then |∂x2 /∂θ| ≤ 1 + c. Note that
∂x1 /∂s = (2d)j is not bounded for d > 1/2, which means that the total width of
the tree is unbounded. However, the condition d > 1/2 ensures the possibility of
gluing together the connectors and the edges of this tree.
Assumptions 4.6. Let V̂ ⊂ Ω̂ × [0, 1] be an inflated vertex of the straightened
tree, where Ŝ0 := Ω̂ × {0}, Ŝj ∼
= k −1/N Ω̂ × {1}, 1 ≤ j ≤ k the corresponding
sections. Let V be the inflated vertex of a given tree TN , and Sj ⊂ ∂V , 0 ≤ j ≤ k
the corresponding sections. We assume that there exists a C 1 -diffeomorphism
F = F (~s, θ) : V̂ → V so that F (Ŝj ) = Sj for 0 ≤ j ≤ k, and such that
¯
¯
¯ ∂F ¯
¯
¯
if θ1 ≤ θ2
¯ ∂θ ¯ < C, and 0 < J(~s, θ1 ) ≤ CJ(~s, θ2 )
hold on V for some C > 0, where J is the Jacobian of F .
Remark 4.7. Assumptions 4.6 imply Assumptions 4.4.
Theorem 4.8. Under Assumptions 4.4 (resp. Assumptions 4.6), the Laplace
operator on TN as defined in Section 2.6, has a purely discrete spectrum.
Proof. Let G : TN → T̂N be the inverse C 1 -mapping of F which is defined in
Assumptions 4.4, and set G(x) := (θ(x), ~s(x)) ∈ T̂N . Denote by J the Jacobian
of F . Let T̂N,j ⊂ TN be the finite subtree
n
o
T̂N,j := (θ, ~s) ∈ T̂N | θ < θj ,
where θj % R̂, and R̂ is the radius of T̂N . Let
³
´
(4.8)
TN,j := F T̂N,j ,
and set
p(x) :=
1
C 2 R̂|R̂ − θ(x)|
.
SCHRÖDINGER OPERATORS ON INFINITE TREES
17
We wish to use Lemma 4.3 with Dj ≡ TN,j . This Lemma requires the compactness
of the identity injection ik : H 1 (Dj ) → L2 (Dj ). Although the boundary of
Dj = TN,j is not C 1 , this injection is still compact. Indeed, the embedding
i : H 1 (D) → L2 (D) is compact for a bounded domain D which has the (inner)
cone property (see [13, Theorem 5.5.1], and the remark on p. 286 therein).
By Lemma 4.3, it is sufficient to prove for the Laplacian that
Z
Z
2
|∇u|2 dx
p(x)|u(x)| dx ≤
(4.9)
TN
T \TN,j
1
for all u ∈ C (TN ) that vanish on the ‘top’ of TN and outside TN,j for some j ≥ 1.
Let u be such a test function, and let v(θ, s) = u(x). Then
¯Z
¯
Z R̂ ¯ ¯2
¯ R̂ ∂v ¯2
¯ ∂v ¯
¯
¯
2
2
¯ ¯ dϑ.
(4.10)
|u(x)| =|v(θ, s)| = ¯
dϑ¯ ≤ |R̂ − θ|
¯ ∂ϑ ¯
¯ θ ∂ϑ ¯
θ
Using the definition of the function p, (4.6), (4.7), (4.10), and Fubini’s theorem,
we obtain,
Z
Z
2
(4.11)
p(x)|u(x)| dx =
p(θ, s)|v(θ, s)|2 Jdξ
TN \TN,j
Z
≤
1
2
TbN \TbN,j C R̂
Z R̂Z
≤
θj
1
Ω̂ C R̂
ÃZ
ÃZ
θ
θ
TbN \TbN,j
!
ÃZ ¯ ¯
¯ ¯2 !
Z R̂Z
R̂ ¯
¯
¯2
1
∂v
∂v
¯ ¯ dϑ Jdξ ≤
¯ ¯ dϑ J(s, θ)dsdθ
¯ ∂ϑ ¯
¯ ∂ϑ ¯
2
θj Ω̂ C R̂
θ
R̂ ¯
!
!
¯ ¯2
Z R̂ÃZ Z R̂ ¯ ¯2
¯
¯ ∂v ¯
∂v
1
¯ ¯ J(s, ϑ) dϑ dsdθ ≤
¯ ¯ J(s, ϑ)dϑds dθ
¯ ∂ϑ ¯
¯ ∂ϑ ¯
C R̂ θj
Ω̂ 0
R̂ ¯
1
≤
C
¯2
Z
Z ¯ ¯2
Z ¯¯X
n
¯
¯ ∂v ¯
∂u
1
∂x
¯
i¯
¯ ¯ Jdξ =
|∇u|2 dx.
¯
¯ dx ≤
¯ ∂ϑ ¯
¯
¯
C
∂x
∂θ
b
i
TN
TN
TN i=1
Since (4.9) is satisfied, the spectrum of the Laplacian on TN is purely discrete. ¤
Remark 4.9. A similar proof applies for a Schrödinger operator on TN with a
bounded from below potential.
4.3. Further results. In this subsection we present two lemmas asserting that
1
(T1 ) and in H01 (TNε ) is conthe L2 -norm of functions which are bounded in H0,ρ
α
centrated on compact sets. These lemmas will be used in Section 6. Their proofs
are similar to those of theorems 4.2 and 4.8, and therefore they are omitted.
Lemma 4.10. Assume that T1 satisfies the assumptions of Theorem 4.2. Suppose
that there exists a weight function 0 < ρ < 1, which is constant on each edge of
T1 , such that ρ ³ ρα and ρ ³ ρβ with a constant c. Denote T1,j = {gen(e) ≤ j}.
(1) Let R(j) be the radius of the maximal (connected) subtree in T \T1,j . Then
Z
Z
c2
2
2
1
|u| ρβ dθ ≤ R(j)
|u0 |2 ρα dθ
∀u ∈ H0,ρ
(T1 ).
α
C
T1 \T1,j
T1
S
ε
ε
(2) Let V := v∈T1 V (v). Then
Z
Z
2
1
|u| ρβ dθ ≤ O(ε)
|u0 |2 ρα dθ
∀u ∈ H0,ρ
(T1 ).
α
V
ε
T1
18
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
Lemma 4.11. Assume that TN satisfies the assumptions of Theorem 4.8.
(1) Let TN,j as defined in (4.8), and let R(j) := R̂ − θj . Then
Z
Z
2
2
2
(4.12)
|u(x)| dx ≤ C R(j)
|∇u|2 dx
∀u ∈ H01 (TN ).
TN \TN,j
S
ε
TN
ε
(2) Let V := v∈T1 V (v). Then
Z
Z
2
(4.13)
|u(x)| dx ≤ O(ε)
ε
TN
Vε
|∇u|2 dx
∀u ∈ H01 (TNε ).
5. Convergence of the spectra of width-weighted operators
In this section we estimate the eigenvalues of the width-weighted operators on
T1 (defined in Section 2.6), for the case where the weight functions and the potential depend on ε, and pointwise converge as ε tends to 0. We treat the weight
functions and potential term as convergent sequences of functions of ε. Hence,
throughout this section we set ε := 1/n, where n ∈ N, and denote the weights
and potentials by ρα,n , ρβ,n and Wn . Accordingly, the corresponding operators are
denoted by Aα,β,n , or An for short. We assume that ρα,n and ρβ,n converge to a
mutual weight function, which we denote by ρ. We denote by W the limit poten1
1
tial of the sequence Wn . We also treat the spaces {H0,n
(T1 )} := {H0,ρ
(T1 )} as
β,n
1
2
∞
a spaces sequence, with a “limiting space” H0,ρ (T1 ). Let {Ln (T1 )}n=1 and L2ρ (T1 )
be the corresponding L2 spaces. Using these notations, we study the asymptotic
behavior of the eigenvalues of An as n → ∞.
Throughout this section we assume that the following conditions are satisfied:
Assumptions 5.1.
(1) T1 has a finite radius.
∞
(2) Assumptions on the weight functions: {ρ1,n }∞
n=1 and {ρ2,n }n=1 are
1
positive bounded weight functions sequences in Lloc (T1 ), such that ρ1,n ³
1
ρ and ρ2,n ³ ρ with the same constant c (so the spaces H0,n
(T1 ) and
1
H0,ρ (T1 ) are equivalent for all n ∈ N). Moreover, for any neighborhood U
containing all the vertices of T1 and a given compact set K b T1 , we have
ρ1,n = ρ2,n = ρ in (T1 ∩ K) \ U for all sufficiently large n.
(3) Assumptions on the potential terms: {WT1 ,n }∞
n=1 is a sequence of real
valued radially symmetric potentials on T1 , for which there exists a positive constant CW such that |WT1 ,n |L∞
≤ CW . Moreover, {WT1 ,n }∞
n=1
ρ (T1 )
converges almost surely (and hence in L1ρ,loc (T1 )) to a potential W , which
satisfies |W |L∞
≤ CW . Without loss of generality, we assume that
ρ (T1 )
WT1 ,n > 1 for all n ∈ N.
Under Assumptions 5.1, we show that the eigenvalues of the operators An converge, as n → ∞, to the eigenvalues of the limit operator A. Here the operators
An are defined by the quadratic forms on H01 (T1 ) × H01 (T1 ):
Z
(5.1)
< An u, φ >n :=
(u0 φ̄0 ρ1,n + WT1 ,n uφ̄ ρ2,n ) dt,
T1
while the limit operator A is defined, similarly, by
Z
(5.2)
< Au, φ >:=
(u0 φ̄0 + W uφ̄)ρ dt.
T1
This result is stated in Corollary 5.4. Notice that since ρ is constant on each edge,
the difference between the derivatives part of A and the Laplacian is manifested
by the Kirchhoff condition.
SCHRÖDINGER OPERATORS ON INFINITE TREES
19
In order to prove the convergence of the spectrum, we need the following lemmas, whose proofs are given later.
Lemma 5.2. For n ∈ N, consider operators An of the form (5.1) which satisfy
1
Assumptions 5.1. Let {un } ⊂ H0,ρ
(T1 ) be a sequence of normalized eigenfunc1
tions of An which converges weakly in H0,ρ
(T1 ) to u. Let λn be the sequence of
corresponding eigenvalues of An . If limn→∞ λn = λ, then Au = λu, and u 6= 0 is
an eigenfunction of the operator A defined in (5.2) with eigenvalue λ. Moreover,
{un } also converges locally uniformly to u.
Lemma 5.3. Consider operators A, and An for n ∈ N, of the form (5.1) and (5.2)
2
respectively which satisfy Assumptions 5.1. Assume also that A −1
n : Lρ (T1 ) →
1
H0,ρ
(T1 ) have uniform bounded norms. Suppose that u 6= 0, and Au = λu in
2
Lρ (T1 ). For each n ∈ N, let wn be the solution of the equation An wn = λu in
L2ρ (T1 ). Then {wn } has subsequence that we continue denoting by {wn }, which
1
converges to u weakly in H0,ρ
(T1 ) and strongly in L2ρ (T1 ). Moreover, {wn } also
converges locally uniformly to u.
∞
∞
Theorem 5.4. Let {ρ1,n }∞
n=1 , {ρ2,n }n=1 and {WT1 ,n }n=1 be sequences of weight
functions and potentials on T1 satisfying Assumptions 5.1. Assume, in addition,
∞
∞
that {WT1 ,n }∞
n=1 are continuous functions and that {ρ1,n }n=1 and {ρ2,n }n=1 equal
ρ except, at most, for O(|ej |/n) neighborhoods of vertices in generation j. Let
a sequence of operators An and a limit operator A be defined by (5.1) and (5.2)
respectively. We denote by λm,n the m-th eigenvalue of An , and by λm the m-th
eigenvalue of A. Then
lim λm,n = λm .
n→∞
Proof. We adapt Attouch’s proof of [2, Theorem 3.71]. Since ρ1,n ³ ρ and ρ2,n ³ ρ
with a positive constant c, and |WT1 ,n | and |W | are bounded by CW , we have for
all u 6= 0 that the Rayleigh quotients satisfy
(5.3)
R
(|u0 |2 ρ1,n + WT1 ,n |u|2 ρ2,n ) dθ
< Au,u >
< An u, u >n
R
= T1
+ 2c2 CW ,
≤ c2
Rn (u) :=
2
< u, u >n
<
u,
u
>
|u|
ρ
dθ
ρ
2,n
T1
and similarly
< An u, u >n
1 < Au, u >
1
≥ 2
− 2 2 CW .
< u, u >n
c < u, u >ρ
c
Fix l ∈ N, by the min-max principle we obtain
1
(λl − 2CW ) ≤ λl,n ≤ c2 (λl + 2CW ),
(5.4)
c2
so, {λl,n } is a bounded sequence. Therefore, there exists a subsequence of {λl,n }
bl ∈ R such that λl,n → λ
bl .
(that we keep denoting by {λl,n }), and λ
bl u
We claim that there exists an eigenfunction u
bl such that Ab
ul = λ
bl , i.e.,
bl } ⊆ {λj }. Indeed, let {ul,n } be the orthonormal sequence of eigenfunctions of
{λ
An that correspond to {λl,n }. We assume that kul,n kn = 1. Then
Z
Z
0
2
| (ul,n ) | ρ1,n dθ =
(λl,n − WT1 ,n )|ul,n|2 ρ2,n dθ ≤ λl,n + CW .
T1
T1
1
It follows that {ul,n } is bounded in H0,ρ
. The weak sequential compactness implies
1
that {ul,n } has a subsequence {ul,n} which converges weakly in H0,ρ
(T1 ). We
b
denote its limit by u
bl . By Lemma 5.2, u
bl 6= 0, Ab
u l = λl u
bl and the convergence is
20
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
bl } ⊆ {λj }. Moreover, (5.4) implies that {λ
bl } is
locally uniform. In particular, {λ
bl } ⊆ {λj }, we have liml→∞ b
an infinite sequence, and since {λ
λl = ∞.
b
Let us now show that {λj } ⊆ {λl }. Assume that there exists an eigenvalue λ
of A such that λ 6= λbl for all l ∈ N, and let u be a corresponding eigenfunction of
A such that ||u||L2ρ = 1.
bm+1 for all limit values b
Take m ∈ N such that λ < λ
λm+1 of the sequence
{λm+1,n }. Set Um,n = span{u1,n , ..., um,n }. By the min-max principle,
λm+1,n = min Rn (v),
⊥
v∈Um,n
⊥
where Rn is defined in (5.3). Therefore, if we could find vn ∈ Um,n
satisfying
lim Rn (vn ) ≤ λ,
n→∞
bm+1 .
then we would arrive to a contradiction of the assumption λ < λ
Let wn be the solutions of the problem An wn = λu. The assumption WT1 ,n > 1
2
implies that A−1
n are uniformly bounded, so {wn } is a bounded sequence in Lρ (T1 ).
1
By Lemma 5.3, up to a subsequence, {wn } converges to u, weakly in H0,ρ
(T1 ),
2
strongly in Lρ (T1 ), and also locally uniformly.
Let us show that limn→∞ Rn (wn ) = λ :
(5.5)
< An wn , wn >n = λ < u, wn >n = λ
Z
[uwn (ρ2,n − ρ) + u(wn − u)ρ] dt + λ.
T1
Since
¯2 ° µ
¯2 ¯Z
¯Z
¶°2
°
°
¯
¯ ¯
¯
¯ uwn (ρ2,n − ρ) dt¯ = ¯ uwn ρ (ρ2,n − ρ) dt¯ ≤ °u ρ2,n − ρ ° kwn k2 2 ,
Lρ
°
° 2
¯
¯ ¯
¯
ρ
ρ
T1
T1
Lρ
Lebesgue’s dominated convergence theorem implies that the first term of the
right-hand side of (5.5) converges to zero, while the second term tends to zero
due to the L2ρ (T1 ) convergence of {wn } to u. Therefore,
(5.6)
lim < An wn , wn >n = λ.
n→∞
Moreover,
(5.7)
< w n , wn > n =
Z
£
T1
¤
(|wn |2 − |u|2 )ρ2,n + |u|2 (ρ2,n − ρ) + |u|2 ρ dt.
The first terms in (5.7) converges to zero due to the strong convergence of wn to
u in L2ρ (T1 ). Indeed,
¯Z
¯
¯ ¡
¯
¢
1/2
1/2
1/2
¯ |wn |2 −|u|2 ρ2,n dt¯ ≤ kwn −uk1/2
L2ρn kwn +ukL2ρn ≤ C kwn −ukL2ρ kwn +ukL2ρ .
¯
¯
T1
The second term in (5.7) converges to zero by Lebesgue’s dominated convergence
theorem. Hence, (5.6) and (5.7) imply that
(5.8)
lim Rn (wn ) = λ.
n→∞
Define
vn := wn −
m
X
k=1
< wn , uk,n >n uk,n .
SCHRÖDINGER OPERATORS ON INFINITE TREES
21
Fix 1 ≤ k ≤ m, and let u
bk be a weak limit of uk,n . It follows (as above) that
(5.9)
lim < wn , uk,n >n = lim < wn , uk,n >ρ =< u, u
bk >ρ .
n→∞
n→∞
By the first part of the proof, u
bk is an eigenfunction of A, and by our assumption,
its eigenvalue is not equal to λ. Therefore, < u, u
bk >n = 0 and by (5.9),
(5.10)
lim < wn , uk,n >n = 0.
n→∞
That implies that {vn } and {wn } share the same L2 -limit u.
Using (5.6) and (5.10), a direct calculation yields that
lim < An vn , vn >n = λ, and
n→∞
lim < vn , vn >n = 1.
n→∞
Hence
lim Rn (vn ) = lim Rn (wn ) = λ.
n→∞
n→∞
By the definition of vn , we have < vn , uk,n >Lnn = 0 for all k = 1, ..., m. Hence, the
bm+1 for some
min-max principle implies that Rn (vn ) ≥ λm+1,n . Therefore, λ ≥ λ
bm+1 }.
limit value b
λm+1 , which contradicts the assumption λ < min{λ
¤
1
1
Remark 5.5. Let {un }∞
n=1 ⊆ H0,ρ (T1 ) ∩ C (T1 ) be a sequence which converges
1
weakly to u in H0,ρ
(T1 ). It follows that {un } is locally a bounded and equicontinuous sequence in C(T1 ). By Arzelà-Ascoli’s theorem, {un }∞
n=1 has a subsequence
that converges locally uniformly to a continuous function u.
Proof of Lemma 5.2. By Remark 5.5, {un } has a subsequence which we continue
denoting by {un }, that converges locally uniformly to u which is continuous on
T1 . We claim: (1) u ∈ Dom(A), (2) Au = λu and , and (3) u 6= 0. The first two
claims follow provided we prove
Z
Z
0 0
(5.11)
(u φ + W uφ)ρ dθ = λ
uφρ dθ
∀φ ∈ C01 (T1 ).
T1
T1
Since {un } are eigenfunctions of An , for each test function φ ∈ C01 (T1 ),
Z
Z
0 0
(5.12)
(un φ ρ1,n + WT1 ,n un φρ2,n ) dθ =
λn un φρ2,n dθ.
T1
T1
1
H0,ρ
By Lebesgue’s theorem applied to ρ1,n and the
bound of un ,
¯Z
¯2
Z
Z
2
¯
¯
0 0
0 2 (ρ − ρ1,n )
¯
¯
|u0n |2 ρ dθ = 0.
(5.13) lim ¯ un φ (ρ − ρ1,n ) dθ¯ ≤ lim |φ |
dθ
n→∞
n→∞
ρ
T1
T1
T1
1
The weak convergence of {un } to u in H0,ρ
(T1 ) implies that
Z
Z
(5.14)
lim
u0n φ0 ρ dθ =
u0 φ0 ρ dθ.
n→∞
T1
T1
By similar arguments, the local uniform convergence of {un } to u, and the a.s.
convergence of WT1 ,n and ρ2,n imply that
(5.15) Z
Z
Z
Z
lim λn un φρ2,n dθ = λ uφρ dθ, and lim WT1 ,n un φρ2,n dθ = W uφρ dθ.
n→∞
T1
T1
n→∞ T
1
T1
Now, (5.12)–(5.15) imply (5.11).
In order to show that u 6= 0, let T1,k = {e ∈ T1 |gen(e) ≤ k}, and let R(k) be the
maximal radius of subtrees in T1 \T1,k . Recall that {un } are eigenfunctions sat1 (T ) = 1, the corresponding eigenvalues sequence {λ n } converges,
isfying kun kH0,ρ
1
22
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
the potential terms {Wn } are bounded by a constant CW for all n ∈ N, and
ρ1,n ³ ρ ³ ρ2,n . Therefore, using (5.13) and the arguments that eigenfunctions
has L2ρ (T1 ) and Hρ1 (T1 ) norms of the same order, we infer that there exist γ, δ > 0
so that, for n large enough, ||un ||L2ρ (T1 ) ≥ γ > 0 and ||u0n||L2ρ (T1 ) ≤ δ.
Therefore, by Lemma 4.10 (1) we have that
Z
Z
Z
2
2
|un | ρ dθ −
|un | ρ dθ =
|un |2 ρ dθ
T1,k
Z
T1
T1 \T1,k
Z
|u0n |2 ρ dθ ≥ γ − δ
c2
|un | ρ dθ − R(k)2
≥
C
T1
2
T1
c2
R(k)2 .
C
Now, choose k large enough such that γ − δ[cR(k)]2 /C > 0. By the local uniform
convergence of un to u, we obtain
Z
Z
Z
c2
2
2
0 < γ − δ R(k) ≤ lim
|un | ρdθ =
|u|2 ρdθ.
|u| ρdθ ≤
n→∞ T
C
T1,k
T1
1,k
Therefore u 6= 0, and u is an eigenfunction of A.
¤
Proof of Lemma 5.3. Since Au = λu and A is invertible, it is sufficient to prove
that Aw = Av (and in particular that w is in the domain of A). But this is
equivalent to
(5.16)
hAw, φi = hAu, φi
1
1
for any function φ in a dense subset of H0,ρ
(T1 ). Recall that w ∈ H0,ρ
(T1 ) and
hAw, φi is defined by (5.2). Let us split the quadratic form (5.2) into
hAw, φi = hAw, φi(1) + hAw, φi(2) ,
where
(5.17)
(1)
hAw, φi
:=
Z
0 0
w φ̄ ρ dt,
T1
(2)
hAw, φi
:=
Z
W w φ̄ρ dt.
T1
Similarly, (5.1) is written as
(2)
hAn w, φin = hAn w, φi(1)
n + hAn w, φin ,
where
(5.18)
hAn w, φi(1)
n
:=
Z
0 0
w φ̄ ρ1,n dt,
T1
hAw, φi(2)
n
:=
Z
W w φ̄ρ2,n dt.
T1
Let Φ(T1 ) be the set of all functions φ ∈ C02 (T1 ) which are constant in some
neighborhood of any vertex v ∈ T1 .
We further observe
0
(1) For any φ ∈ Φ(T1 ) and sufficiently large n, φ = 0 whenever ρ1,n 6= ρ or
ρ2,n 6= ρ by Assumption 5.1 (2). Hence, for a given φ ∈ Φ(T1 )
(1)
hAn wn , φi(1)
n = hAwn , φi .
for all sufficiently large n.
1
(2) Since w is the weak limit of wn in H0,ρ
(T1 ) it follows by (1) that
(1)
lim hAn wn , φi(1)
= hAw, φi(1) .
n = lim hAwn , φi
n→∞
n→∞
SCHRÖDINGER OPERATORS ON INFINITE TREES
23
(3) By Assumption 5.1 and the strong convergence of wn to w in L2ρ (T1 ) we
obtain
(2)
lim hAn wn , φi(2)
.
n = hAw, φi
n→∞
(4) By (2) and (3) we obtain
lim hAn wn , φin = hAw, φi .
n→∞
for any φ ∈ Φ(T1 ).
(5) Since An wn = λu = Au by assumption we obtain hAn wn , φin = λ hu, φin =
hAu, φin . Since ρn,1 → ρ in measure, it follows that
hAw, φi = lim hAn wn , φin = λ hu, φiρ = hAu, φi .
n→∞
So, (5.16) is proved for any φ ∈ Φ(T1 ). The proof is completed by observing that
1
Φ(T1 ) is clearly dense in H0,ρ
(T1 ).
¤
6. ε-dependent bounds for the eigenvalues of N -dimensional tree
In this section, we consider the spectrum of the Schrödinger operator
Lε := −∆ + WTNε
on TNε , where N is dimension of the tree, and WTNε is a continuous bounded
potential on TNε . Without loss of generality, we assume that WTNε ≥ 0.
We prove that the eigenvalues of Lε are bounded from above and below by
functions φεQ , φεP of the eigenvalues of weighted operators Aε on T1 of the form
(6.1)
1 d
Aε := −
ρb,ε dθ
µ
d
ρa,ε
dθ
¶
+ WT1 ,ε ,
for a suitable choice of weight functions ρa,ε , ρb,ε and a potential WT1 ,ε on T1 of
the form
 R
W ε (θ, s) ds
ε

 Ωεe TN
θ ∈ E (e),
ε
|Ωe |
(6.2)
WT1 ,ε (θ) :=

 P
ε
e∈N (v) be ψ(e) (θ) θ ∈ V (v),
R
ε
ε
where be = (|Ωεe |)−1 Ωε WTNε (pεe , s) ds, pεe = ∂V (v) ∩ e is the end point of V (v)
e
corresponding to e ∈ N (v), and {ψ(e) } is the partition of unity in a neighborhood
of the vertex v defined in Section 3. The functions φεQ and φεP converge to the
identity function as ε tends to zero.
6.1. Rayleigh quotients of Schrödinger operator on T1 and TNε . The comparison between the Rayleigh quotients on T1 and TNε involves the construction of
1
ε
1
1
ε
: H01 (TNε ) → H0,ρ
transformations Qε : H0,ρ
∗ (T1 ), where
∗ (T1 ) → H0 (TN ) and P
(N
−1)
ρ∗ : T1 → R is defined by ρ∗ (θ) := δe
|Ωe | for θ ∈ e. We devote the following
two subsections for the definitions of these transformations.
24
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
1
ε
1
1
6.1.1. The mapping Qε : H0,ρ
∗ (T1 ) → H0 (TN ). Given a function f ∈ Hρ∗ (T1 ) and
a vertex v, we denote by feε = f (pεe ) and f~ε = {feε }e∈N (v) . Qε (f ) is defined as
follows:
½
f (θ)
x = (θ, s) ∈ E ε ,
ε
(6.3)
Q (f )(x) = P
ε ε
ε
e∈N (v) fe φ(e) x ∈ V (v),
where {φε(e) } is a partition of unity of V ε (v) as defined in Section 3. We denote
Q(f ) := Q1 (f ). We also define
( ∗
ε
ρ n
θ ∈ E (e),
o
ε
B
A
ε
(6.4)
ρQ (θ) =
max αβ A , αβ B ρ∗ θ ∈ V (v),
where αA , β A , αB and β B are defined in Section 3 and Lemma 3.1.
1
Lemma 6.1. There exists c > 0 such that for any f ∈ H0,ρ
∗ (T1 ) and 0 < ε < 1,
we have
(1) Qε (f ) ∈ L2 (TNε ).
R
R
0
(2) T ε |∇Qε (f )|2 dx ≤ ε(N −1) T1 |f |2 ρεQ dθ . Moreover, Qε (f ) ∈ H01 (TNε ).
N
R
R
0
(3) T ε |Qε (f )|2 dx ≥ ε(N −1) T1(|f |2 −cε|f |2 )ρ∗ dθ.
N
R
R
0
(4) T ε WTnε |Qε (f )|2 dx ≤ ε(N −1) T1 (WT1 ,ε |f |2 + O(ε)|f |2 )ρ∗ dθ.
N
ε
Proof. 1. We denote a normalized connector V := (εδ)−1 V (v), where δ = δv =
δ gen(v) corresponds to the vertex v in question, and
µ ¶
µ ¶
θ
θ
∗
∗
b
(6.5)
f (θ) := f
,
ρb (θ) := ρ
εδ
εδ
are the representation of f and ρ∗ in V (here θ = 0 corresponds to v).
Z Z
Z
Z
Z
ε
2
2
(N −1)
ε
2
(N −1)
|Q (f )| dx =
|f (θ)| dsdθ = (εδ)
|Ω ||f | dθ = ε
|f |2 ρ∗ dθ.
Eε
E
ε
Ωε
E
ε
E
ε
ε
For the connector V , we have by Lemma 3.1 that
Z
Z
ε
2
N
|Q (f )| dx == (εδ) |Q(f )|2 dx ≤ (εδ)N αB |f~|2 .
Vε
V
By Lemma 3.3 and since δ < 1, we have that
Z
Z
ε(N −1) αB
(εδ)N αB
0 2
2 ∗
N B ~2
b
b
(|f | + |f | )b
ρ dθ ≤
(εδ) α |f | ≤
(|εf 0 |2 + |f |2 )ρ∗ dθ.
(N
−1)
B
B
ε
δ
β V
β
V
In particular, we proved
kQε (f )k2L2 (TNε ) ≤ ε(N +1) kf 0 k2L2ε (T1 ) + ε(N −1) kf k2L2ε (T1 ) .
(6.6)
2.Z
ρ
ε
Eε
2
|∇Q (f )| dx =
ε
Z
0
Eε
2
|f | dx =
Z
E
ε
Z
ρ
Q
0
Ωε
2
Q
(N −1)
|f | dsdθ = ε
Z
E
0
ε
|f |2 ρ∗ dθ.
For V we use similar considerations to those used in part 1:
Z
Z
ε
2
(N −2)
|∇Q(f )|2 dx = (εδ)(N −2) f~Af~∗
|∇Q (f )| dx = (εδ)
ε
V
V
(N −2) A Z
AZ
(εδ)
α
(N −2) A ~ ~ 2
0 2 ∗
(N −1) α
b
≤ (εδ)
α |f x1| ≤ (N −1) A |f | ρ dθ = ε
|f 0 |2 ρ∗ dθ.
A
ε
δ
β V
β V
SCHRÖDINGER OPERATORS ON INFINITE TREES
So, Qε (f ) ∈ H01 (TNε ) by definition (6.4).
3. By Lemma 4.10,
Z
Z
Z
ε
2
ε
2
(N −1)
|Q (f )| dx ≥
|Q (f )| dx = ε
ε
TN
=ε
Z
ε \∪ V ε (v)
TN
v
(N −1)
2 ∗
T1
Z
2 ∗
(N −1)
|f | ρ dθ−ε
ε
Z
(N −1)
|f | ρ dθ ≥ ε
∪v V (v)
∪e E(e)
25
|f |2 ρ∗ dθ
2 ∗
Z
N
|f | ρ dθ−cε
T1
The proof of (4) is a simple extension of (6.6).
|f 0 |2 ρ∗ dθ.
T1
¤
1
Corollary 6.2. There exists a constant c > 0 such that for all f ∈ H0,ρ
∗ (T1 ) and
0 < ε sufficiently small, the Rayleigh quotients
R
ε
2
ε
2
ε |Q (f )| ) dx
ε (|∇(Q f )| +WTn
TN
ε
R
,
RH01 (TNε ) [Q f ] :=
|Qε f |2 dx
Tε
N
and
1
RH0,ρ
[f ] :=
∗ (T1 )
R
T1
satisfy the inequality
RH01 (TNε ) [Qε f ] ≤
(6.7)
(|fθ |2 ρεQ +WT1 ,ε |f |2 ρ∗ ) dθ
R
|f |2 ρ∗ dθ
T1
1
(1 + O(ε))RH0,ρ
[f ]
∗ (T1 )
1
1 − cεRH0,ρ
[f ]
∗ (T1 )
.
1
Remark 6.3. Notice that RH0,ρ
[f ] depends on ε and is the Rayleigh quotient
∗ (T1 )
of the width-weighted operator Aε defined in (6.1), substituting ρα,ε = ρεQ and
ρb,ε = ρ∗ .
1
1
ε
6.1.2. The mapping P ε : H01 (TNε ) → H0,ρ
∗ (T1 ). Given a function u ∈ H0 (TN ), a
vertex v and edges e ∈ N (v), we denote
Z
1
u(pεe , s) ds,
~u = ~uv := {ue }e∈V (v) ,
ue := ε
|Ωe | Ωεe
ε
ε
where pεe = ∂V (v) ∩ e are the end points of V (v). Define
 R
u(θ, s)ds

ε
Ωε


θ∈E ,

ε
|Ω |
(6.8)
P ε (u)(θ) :=
X
ε
ε


ue ψ(e)
(θ) θ ∈ V ,


e∈V (v)
where {ψ(e) } is the partition of unity in a neighborhood of the vertex v defined
in Section 3. We also define
( ∗
ε
ρ n
θ∈E ,
o
ε
B
A
(6.9)
ρP (θ) :=
ε
min αβ A , αβ B ρ∗ θ ∈ V .
Lemma 6.4. There exists c > such that for any u ∈ H01 (TNε ) and 0 < ε sufficiently small, we have
(1) P ε (u) ∈ L2ρ∗ (T1 ).
R
R
0
1
(2) ε(N −1) T1 |(P ε u) |2 ρεP dθ ≤ T ε |∇u|2 dx. In particular, P ε (u) ∈ H0,ρ
∗ (T1 ).
N
26
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
(3) ε(N −1)
(4) ε(N −1)
R
R
T1
|P ε u|2 ρ∗ dθ ≥
R
ε
TN
[(1 −
WT1 ,ε |P ε u|2 ρ∗ dθ ≤
T1
R
ε
TN
√
ε)|u|2 − cε|∇u|2 ] dx.
√
(1 + 2 ε)(WTNε |u|2 + O(ε)|∇u|2) dx.
Proof. Throughout the proof we denote u
b(x) = u (δεx) for x ∈ V (so, εδx ∈ V ε ).
Similarly, u
e(θ, s) = u (θ, εδs) for (θ, s) ∈ E × Ω (so, (θ, εδs) ∈ E ε ).
1. For each edge E ε ,
¯
¯2
Z
Z
Z
Z
¯
¯
(N −1)
ε 2 ∗
ε
ε 2
ε ¯ 1
ε
|P u| ρ dθ =
|Ω ||P u| dθ =
|Ω | ¯ ε
u(θ, s)ds¯¯ dθ
ε
ε
ε
|Ω | Ωε
E
E
E
Z Z
Z
2
≤
|u(θ, s)|2 dx.
|u(θ, s)| dsdθ =
E
ε
Eε
Ωε
Using Lemma 3.1, we obtain for the connector V ε
Z
Z
X
(N −1)
ε 2 ∗
N
ε
|P u| ρ dθ = ε δ
ue uẽ ψ(e) ψ(ẽ) ρ∗ dθ = εN δ~uB(~u)∗ ≤ (εδ)N αB |~u|2 .
V
ε
V
e,ẽ∈N (v)
By Lemma 3.5, and assuming δ < 1, we obtain
Z
B Z
αB
N B
2
Nα
2
2
(εδ) α |~u| ≤ (εδ) B
(|∇b
u| + |b
u| ) dx ≤ B
(|ε∇u|2 + |u|2 ) dx.
β V
β
ε
V
2. For an edge E ε we have
Z
Z ¯
¯
¯ ε 0 ¯2 ∗
(N −1)
(N −1)
(6.10) ε
¯(P u) ¯ ρ dθ = ε
E
ε
E
=
For V ε , we have
Z
(N −1)
(6.11) ε
V
Z
ε −1
E
ε
|Ω |
¯
Z
¯ 1
¯
ε ¯ |Ωε |
µZ
Ωε
¯2
∂u(θ, s) ¯¯ ∗
ds¯ ρ dθ
∂θ
Ωε
¶2
Z
∂u(θ, s)
ds dθ ≤ |∇u|2 dx.
∂θ
Eε
by Lemma 3.1 and (6.8)
Z
¯
¯
X
¯ ε 0 ¯2 ∗
(N −1)
ε 0
ε 0 ∗
ue u(e)
(ψ(e)
) (ψ(e)
¯(P u) ¯ ρ dθ = ε
˜
˜ ) ρ dθ
ε
ε
V
e,ẽ∈N (v)
= ε(N −2) δ −1~uA(~u)∗ ≤ (εδ)(N −2) αA |~ux~1|2
(N −2) α
≤ (εδ)
ε
A
βA
Z
αA
|∇b
u| dx = A
β
2
V
Z
V
ε
|∇u|2 dx.
3. In the edges E , we use the same argument as in [21]. By the inequality
√
√
(6.12)
(a + b)2 ≥ (1 − ε)a2 − b2 / ε,
we have that
Z
Z
Z
(N −1)
ε 2 ∗
ε 2
ε
|P u| ρ dθ =
|P u| ds dθ =
|u + (P ε u − u)|2 ds dθ
ε
ε
ε
E
E
E
¸
Z Z ·
√
1
2
ε
2
≥
(1 − ε)|u(θ, s)| − √ |P u − u(θ, s)| dsdθ
ε
ε
E
Ωε
¸
Z Z ·
√
1
2
(N −1)
2
e−u
e(θ, s)| dsdθ.
=ε
(1 − ε)|e
u(θ, s| ) − √ |P u
ε
ε
E
Ω
SCHRÖDINGER OPERATORS ON INFINITE TREES
27
Notice that for each θ we have that P u
e(θ) − u
e(θ, s) has average zero on Ω.
1
inequality
RBy Poincaré
R in 2H (Ω), there exists a constant D > 0 such that
2
|P
u
e
−
u
e
|
ds
≤
D
|∇e
u| ds and hence,
Ω
Ω
¸
Z Z·
Z
√
1
2
ε 2 ∗
(N −1)
2
(N −1)
u(θ, s)| − √ D|∇e
|P u| ρ dθ ≥ ε
(1− ε)|e
u(θ, s)| dsdθ
ε
ε
ε
ε
E
E Ω
Z
£
¤
√
=
(1 − ε)|u|2 − ε3/2 D|∇u|2 dx.
Eε
Therefore,
Z
∪e
E ε (e)
√
[(1 − ε)|u|2 − ε3/2 D|∇u|2 ] dx ≤ ε(N −1)
On the other hand, by Lemma 4.11,
Z
Z
√
2
3/2
2
[(1− ε)|u| −ε D|∇u| ]dx ≤
∪v V
ε (v)
∪v V
Z
T1
ρ∗ |P ε u|2 dθ.
√
√
(1− ε)|u|2 dx ≤ cε(1− ε)
ε (v)
Z
|∇u|2 dx.
ε
TN
Summing the last two inequalities, we obtain the proof of part 3.
4. Since
Z
Z
(N −1)
ε 2 ∗
ε
WT1 ,ε |P u| ρ dθ =
WTNε |P ε u|2 ds dθ,
E
ε
Eε
it is sufficient to prove for the edges that
Z
Z
√
ε
2
WTNε |P u| ds dθ ≤
(1 + 2 ε)WTNε |u|2 + O(ε)|∇u|2) dx.
Eε
Using (6.12), we have that
Z
Z
√
2
ε
WTN |u| dx ≥ (1 − ε)
Eε
Eε
W
Eε
ε
TN
1
|P u| ds dθ − √
ε
ε
2
Z
Eε
Therefore, if 0 < ε < 1 is small enough so that 1 ≤ (1 −
Poincaré inequality, there exists a constant D such that
WTNε |u − P ε u|2 ds dθ.
√
√
ε)(1 + 2 ε), then by
(6.13)
½Z
¾
Z
Z
√
1
ε
2
2
ε
2
WTNε |P u| ds dθ ≤ (1 + 2 ε)
WTNε |u| dx + √
WT ε |u−P u| ds dθ
ε Eε N
Eε
Eε
Z
Z Z
√
√ 1
2
≤ (1 + 2 ε)
WTNε |u| dx + CW (1 + 2 ε) √
|e
u − Pu
e|2 (εδ)(N −1) ds dθ
ε
ε
ε
E
E
Ω
Z
Z
√
√ (εδ)2
2
≤ (1 + 2 ε)
WTNε |u| dx + CW D(1 + 2 ε) √
|∇s u|2 ds dθ
ε Eε
Eε
Z
Z
√
2
3/2
WTNε |u| dx + O(ε )
|∇u|2 dx.
≤ (1 + 2 ε)
Eε
Eε
For the connectors we obtain by Lemma 4.10 and part 2,
Z
Z
Z ¯
¯
¯ ε 0 ¯2 ∗
(N −1)
ε 2 ∗
N
ε
WT1 ,ε |P u| ρ dθ ≤ ε CW
¯(P u) ¯ ρ dθ ≤ εCW
ε
∪v V (v)
T1
which, together with (6.13), yields the proof of part 4.
ε
TN
|∇u|2 dx
¤
28
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
Corollary 6.5. For all ε > 0 sufficiently small, there exists a constant c > 0
such that the Rayleigh quotients
R ¡ ε 02 ε
¢
|(P u) | ρP + WT1 ,ε |P ε u|2 ρ∗ dθ
T1
ε
R
1
RH0,ρ
[P u] :=
,
∗ (T1 )
|u|2 ρ∗ dθ
T1
and
RH01 (TNε ) [u] :=
satisfy
(6.14)
R
ε
TN
(|∇u|2 + WTnε |u|2 )dx
R
|u|2 dx
Tε
N
√
ε)]RH01 (TNε ) [u]
[1
+
O(
ε
√
1
RH0,ρ
[P
u]
≤
(T
)
1
∗
1 − ε − cεRH01 (TNε ) [u]
∀u ∈ H01 (TNε ).
1
Remark 6.6. Notice that RH0,ρ
[P ε u] is the Rayleigh quotient of the width∗ (T1 )
weighted operator Aε defined in (6.1), substituting ρα,ε = ρεP and ρb,ε = ρ∗ .
6.2. T1 -based estimates for the spectrum on TNε . Rubinstein and Schatzman
have proved the following general lemma [21].
Lemma 6.7. Let Aj be bounded below, selfadjoint operators defined on Hilbert
spaces Hj , where j = 0, 1, and let {λm (Aj )} be the nondecreasing sequence of the
corresponding eigenvalues. Denote by Dj the domain of the maximal quadratic
form associated with Aj and by Rj the Rayleigh quotient associated with Aj .
Suppose that there exists a continuous linear operator S mapping D 1 to D0 and
an increasing function φ : R → R ∪ {+∞} such that exp(−φ) is continuous, and
R0 (Su) ≤ φ(R1 (u))
∀u ∈ D1 \ ker(S).
Assume that for a given m,
(6.15)
µ := inf{R1 (v) | v ∈ D1 ∩ ker(S), v 6= 0} > λm (A1 ).
Then
(6.16)
λm (A0 ) ≤ φ(λm (A1 )).
ε
Using Lemma 6.7, we obtain bounds for the eigenvalues of TNε . Let νm
denotes
the m-th eigenvalue of the Schrödinger operator
Lε := −∆ + WTNε .
Denote the operators
µ
¶
1 d
ε
ε d
AQ := − ∗
ρQ
+ WT1 ,ε ,
ρ dθ
dθ
AεP
1 d
:= − ∗
ρ dθ
µ
ρεP
d
dθ
¶
+ WT1 ,ε ,
and let µεm (resp. λεm ) be the m-th eigenvalue of AεQ (resp. AεP ). We will omit
ε
the superscript ε in νm
, µεm , and λεm whenever there is no danger of confusion.
Theorem 6.8. Using the notations above, for all M ∈ N there exist εM > 0 and
a constant c > 0 such that for all m ≤ M and 0 < ε < εM , we have
(6.17)
ε
νm
≤ φεQ (µεm ),
and
(6.18)
ε
λεm ≤ φεP (νm
),
SCHRÖDINGER OPERATORS ON INFINITE TREES
where

 (1+cε)x x < (cε)−1 ,
ε
φQ (x) :=
1−cεx

+∞
otherwise,
29

√
1− ε
 (1+cε)x
√
x<
,
and φεP (x) := 1− ε−cεx
cε

+∞
otherwise.
Proof of Theorem 6.8. Without loss of generality, we assume that WT1 is positive.
In order to prove (6.17), we wish to apply Lemma 6.7 on S = Qε , D0 = H01 (TNε ),
1
ε
D1 = H0,ρ
∗ (T1 ), A0 = Lε , A1 = AQ , R0 = RH 1 (T ε ) , and R1 = RH 1
(T1 ) . We,
0
N
0,ρ∗
therefore, show that there exists C > 0 such that for any ε > 0
n
o
1
ε
1
(6.19)
inf RH0,ρ
[f
]
|
f
∈
ker
Q
,
f
=
6
0
≥
.
∗ (T1 )
Cε2
Indeed,
©
εª
1
∀θ ∈ T1 \ ∪v V .
ker Qε = f ∈ H0,ρ
∗ (T1 ) | f (θ) = 0
1
Therefore, in order to estimate RH0,ρ
[f ] for f ∈ ker(Qε ), we actually need to
∗ (T1 )
ε
estimate this quotient in each component V ∩ e. However, we have that
¯Z θ
¯2
Z θ
¯
¯
0
0
ε
2
¯
¯
|f (θ)| = ¯
f dϑ¯ ≤ |p − θ|
|f |2 dϑ,
pε
pε
ε
where pε ∈ ∂V . Multiply the above by ρ∗ (which, we recall, is constant on each
ε
component V ∩ e), we find the existence of C > 0 such that
µ
¶
Z
Z
Z θ
Z
0
0 2 ∗
2 ∗
ε
2
|f | ρ dθ ≤
|p − θ|
|f |2 ρ∗ dθ.
|f | ρ dϑ dθ ≤ Cε
ε
V ∩e
ε
pε
V ∩e
V ∩e
Thus, (6.19) is verified provided ε is sufficiently large. Hence, (6.17) follows from
Corollary 6.2 and Lemma 6.7.
1
In order to prove (6.18), we wish to apply Lemma 6.7 to S = P ε , D0 = H0,ρ
∗ (T1 ),
1
ε
1
D1 = H01 (TNε ), A0 = AεP , A1 = Lε , R0 = RH0,ρ
,
and
R
=
R
.
To
this
1
H0 (TN )
∗ (T1 )
end, we show that there exists C > 0 such that for any ε > 0
o
n
1
(6.20)
inf RH01 (TNε ) [u] | u ∈ ker P ε , u 6= 0 ≥
.
Cε
We notice that if u ∈ ker P ε , then its averages on the cross sections Ωεj of Ejε
vanish. Therefore, using the (N −1)-dimensional Poincaré inequality for functions
whose average is zero, we obtain that there is a constant D such that:
(6.21)
Z
Z Z
Z
|u|2 ds dθ ≤ Dε2(N −1)
Eε
E
ε
|∇s u|2 ds dθ ≤ Dε2(N −1)
Ωε
(|∇u|2 + WTNε |u|2 ) ds dθ.
Eε
By Lemma 4.11, there exists C > 0 such that for any u ∈ H01 (TNε )
Z
Z
2
(6.22)
|u| dx ≤ Cε
(|∇u|2 + WTNε |u|2 ) dx.
∪v V ε (v)
ε
TN
Therefore, (6.21) and (6.22) imply (6.20). Thus, (6.18) follows by Corollary 6.5
and Lemma 6.7.
¤
Remark 6.9. Theorem 6.8 is similar to [21, Theorem5] proved for a finite graph
with a constant-width thin domain.
Theorem 6.10. For each m ∈ N, the m-the eigenvalue of the Schrödinger operator Lε on H01 (TNε ) converges as ε → 0 to the m-the eigenvalue of limit widthweighted operator A on H01 (T1 ).
30
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
Proof. We use in this proof the notations of Theorem 6.8. Notice that for small
enough ε, φεQ and φεP are continuous monotone increasing function, which satisfy
lim φεQ (x) = x, lim φεP (x) = x.
ε→0
ε→0
Moreover, since the operators we refer to in Theorem 6.8 satisfy the conditions
of Theorem 5.4, we have for each m ∈ N that both µεm and λεm (see (6.17) and
(6.18)) converge as ε → 0 to the m-th eigenvalue of the limit width-weighted
operator A. Since A has a discrete spectrum, the result follows.
¤
7. Convergence of eigenfunctions of Laplace operator on TNε
In [7, 8], Kosugi has proved that the solutions of ∆u+f (u) = 0 in thin networkshaped bounded domains that satisfy Neumann boundary condition, converge to
solutions of appropriate equations on the skeleton of the domain. In [7], Kosugi
deals only with domains which are formed by joining straight tubes around some
graph, while in [8] the results are extended to general domains around graphs.
However, trees with infinite number of vertices and nonsmooth boundaries are not
considered in these papers. Using the transformation P ε developed for Theorem
1
6.5, we give a simple proof for the convergence of projections into H0,∗
(T1 ) of
1
ε
eigenfunctions uε of the Laplace operator on H0 (TN ). Specifically, we show in
Theorem 7.2 that P ε uε converges to eigenfunctions of the following limit widthweighted operator on T1
³ 0 ´0
L∗ u := (ρ∗ )−1 ρ∗ u .
First, we need to prove the following auxiliary Lemma.
Lemma 7.1. Assume that u ∈ H01 (Tnε ) satisfies ||u||H01 (Tnε ) = ε(n−1)/2 . Fix a vertex
ε
ε
v, and denote by pe the ‘end point’ in V ∩ E (e). Then there is a constant C
which depends on v but is independent on ε such that for e, ẽ ∈ N (v) we have
p
(7.1)
|P ε u(pe ) − P ε u(pẽ )| ≤ C dist(pe , pẽ ) ,
where dist(·, ·) is the standard distance function on T1 .
Proof. Notice that since C 1 (Tnε ) is dense in H01 (Tnε ) we may assume without
loss of generality that u ∈ C 1 (Tnε ).
ε
Let q, r ∈ V ∩ e. By (6.11),
¯2
¯
¯Z q
Z q¯
¯
¯ d ε ¯2 ∗
¯
d
1
ε
ε
ε
2
¯ (P u)¯ ρ dθ
(P u)dθ¯¯ ≤ dist(q, r) ∗
|P u(q) − P u(r)| = ¯¯
¯
ρe r ¯ dθ
r dθ
Z
ε1−n αA
ε1−n αA
≤ dist(q, r) ∗ A
|∇u|2 dsdθ ≤ dist(q, r) ∗ A εn−1 ≤ Cdist(q, r)
ρe β V ε
ρe β
p
for some constant C. Therefore, |P ε u(pe ) − P ε u(pẽ )| ≤ 2C dist(pe , pẽ ).
¤
Theorem 7.2. Let uε ∈ H01 (TNε ) be an eigenfunction with eigenvalue λε of the
Laplace operator on TNε , such that kuε kL2 (TNε ) = ε(N −1)/2 . Assume that limε→0 λε =
λ∗ . Then there exists an eigenfunction u∗ of L∗ which corresponds to λ∗ , such that
up to a subsequence,
u∗ = lim P ε uε
ε→0
locally uniformly.
SCHRÖDINGER OPERATORS ON INFINITE TREES
31
Proof. By elliptic regularity, uε ∈ C 2 (TNε ). Our proof consists of three steps.
2
Step 1. Let us show that P ε uε converges to a solution u∗ of ddθu2 = λ∗ u on each
edge of e ∈ T1 .
By parts 2 and 4 of Lemma 6.4 (with W = 1), we obtain that P ε uε are uniformly
bounded in H∗,1 (T1 ). This implies, in particular, that P ε uε are uniformly locally
bounded in L∞ (T1 ). In addition, (up to a subsequence) limε→0 P ε uε = u∗ holds
locally uniformly by Arzelà-Ascoli’s Theorem. Fix an edge e ∈ T1 , and θ1 , θ2 ∈ e.
ε
Let ζ(θ) ∈ C0∞ ([θ1 , θ2 ]). If ε > 0 is sufficiently small, then θ1 , θ2 ∈ E . Therefore,
Z
θ2
ε
00
P uε (θ)ζ (θ)dθ =
θ1
1
= ε
|Ω |
Z
Ωε
Z
Z
θ2
θ1
1
|Ωε |
µZ
Ωε
¶
uε (θ, s)ds ζ 00 (θ) dθ
Z Z θ2
1
uε (θ, s)∆ζ(θ) dθ ds = − ε
∇uε (θ, s) · ∇ζ(θ) dθ ds
|Ω | Ωε θ1
θ1
Z Z θ2
Z θ2
λε
=− ε
uε (θ, s)ζ(θ) dθ ds = −λε
P ε uε (θ)ζ(θ) dθ.
|Ω | Ωε θ1
θ1
θ2
Hence, P ε uε ∈ H 2 ([θ1 , θ2 ]) and −(P ε uε )00 = λε P ε uε in the weak sense and by
ε
elliptic regularity also in the strong sense. Moreover P ε uε is C ∞ in E . Since
λε → λ∗ and P ε uε → u∗ uniformly on e, the second derivatives (P ε uε )00 converge uniformly to (u∗ )00 , which also implies the same convergence for the first
derivatives (P ε uε )0 .
Step 2. We show now that u∗ is in the domain of L∗ . For this, we must only
show that u∗ satisfies the corresponding Kirchhoff’s conditions. The continuity
at the vertices is satisfied by Lemma 7.1. The second Kirchhoff condition is given
by
X
ρ∗e u0e (v) = 0,
e∈N (v)
where N (v) is the set of all edges adjacent to the vertex v. Recall that ρ∗e =
(N −1)
δe
|Ωe | takes a constant value on each edge e.
Let U ⊂ T1 be a neighborhood of the vertex v which contains no other vertex,
and let θe ∈ ∂U be the point of ∂U contained in e ∈ N (v). Let U ε ⊂ TNε be the
inflation of U , that is, U = U ε ∩ T1 . In particular, for sufficiently small ε we have
∂U ε = (U ε ∩ ∂TNε )
[
Se ,
e∈N (v)
where Se = {s; (θe , s) ∈ E ε (e)}. Let ζε ∈ C ∞ (U ε ) be a function which does not
depend on s in the edges, satisfies ζε (x) = 1 for all x ∈ V ε (v), 0 ≤ ζε (x) ≤ 1 for
all x ∈ U ε , and vanishes around each Se . Since uε is an eigenfunction, we have
λε
Z
uε ζε dx =
Uε
Z
V ε (v)
∇uε · ∇ζε dx +
Z
U ε (v)\V ε (v)
∇uε · ∇ζε dx
Z
∂uε dζε
=
ds dθ.
U ε (v)\V ε (v) ∂θ dθ
32
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
As ζε depends only on θ on U ε (v)\V ε (v) and equals one at pe , we get
Z θe
Z
X
∂P ε uε dζε
∂uε dζε
ε
|Ωe |
dsdθ =
dθ
(7.2)
∂θ dθ
p
U ε (v)\V ε (v) ∂θ dθ
e
e∈N (v)
·
¸
Z θe
X
∂P ε uε
ε
ε
00
(pe )ζε (pe ) +
=−
|Ωe |
(P uε ) ζε dθ
∂θ
p
e
e∈N (v)
¸
· ε
Z θe
X
∂P uε
ε
ε
P uε ζε dθ
(pe ) − λε
=−
|Ωe |
∂θ
pe
e∈N (v)
Z
ε
X
ε ∂P uε
=−
|Ωe |
(pe ) + λε
uε ζε dx.
∂θ
U ε (v)\V ε (v)
e∈N (v)
The change of order of integration and differentiation in the first line of (7.2) is
easily justified by approximating uε with a smooth function. We therefore obtain
that
Z
ε
X
ε ∂P uε
(pe ) = −λε
uε ζε dx,
|Ωe |
∂θ
V ε (v)
e∈N (v)
ε
and since |V (v)| = cεN , we arrive at the estimate
¯
¯
¯
¯
µZ
¶1/2
ε
¯
¯ X ∗ ∂P uε
1−N
N/2
2
¯
(7.3)
ε λε
uε (x) dx
(pe )¯¯ ≤ cε
ρe
¯
∂θ
V ε (v)
¯
¯
e∈N (v)
= cλε ε1/2 .
Letting ε → 0, we obtain by Step 1 that the left hand side of (7.3) converges to
¯
¯
¯
¯ X
¯
¯
∗ ∗ 0
¯
¯
ρ
(u
)
(v)
e
e
¯
¯
¯
¯e∈N (v)
and the right hand side to zero.
Step 3. It remains to prove that u∗ 6≡ 0. Let TN,j denote the j first generations
in TN . By lemmas 6.4 and 4.10 there are constants c, C > 0 and a function R(j)
which tends to zero as j → ∞ such that
Z
(N −1)
ε
|P ε uε |2 ρ∗ dθ
T1,j
Z
Z
ε
2 ∗
(N −1)
(N −1)
|P ε uε |2 ρ∗ dθ
|P uε | ρ dθ − ε
=ε
T1 \T1,j
T1
2
√ (N −1)
c
ε)ε
− cλε εN − 2 R(j)2 λε ε(N −1)
·C
¸
√
c2
(N −1)
2
(1 − ε) − cλε ε − 2 R(j) λε .
=ε
C
√
2
Choose ε > 0 small enough and jR large enough so that (1− ε)−cλε ε− Cc 2 R(j)2 λε≥
γ for a constant γ > 0. Then T1,j |P ε uε |2 ρ∗ dθ ≥ γ > 0. By the local uniform
convergence of P ε uε to u∗ we have that
Z
Z
Z
∗ 2 ∗
∗ 2 ∗
|u | ρ dθ ≥
|u | ρ dθ = lim
|P ε uε |2 ρ∗ dθ ≥ γ > 0,
≥ (1 −
T1
T1,j
ε→0
T1,j
SCHRÖDINGER OPERATORS ON INFINITE TREES
so, u∗ 6≡ 0.
33
¤
Acknowledgments
The paper is based on part of the Ph. D. thesis [25] of Daphne Zelig, completed
in 2005 at the Technion, under the supervision of Moshe Israeli, Yehuda Pinchover and Gershon Wolansky. The authors would like to thank Professors Peter
Kuchment, Alexander Sobolev, and Michael Solomyak for valuable discussions.
This work was partially supported by the RTN network “Nonlinear Partial Differential Equations Describing Front Propagation and Other Singular Phenomena”, HPRN-CT-2002-00274. The works of Y. P. and G. W. were also partially
supported by the Israel Science Foundation (grants 1136/04 and 406/05, respect.)
founded by the Israeli Academy of Sciences and Humanities, and by the Fund for
the Promotion of Research at the Technion.
References
[1] D. E. Anagnostou, M. T. Chryssomallis, J. C. Lyke, and C. G. Christodoulou, Improved multiband performance with self-similar fractal antennas, in “IEEE ASP
Tropical Conference on Wireless Communications Technology”, pp. 271–272, Honolulu, 2003.
[2] H. Attouch, “Variational Convergence for Functions and Operators”, Applicable
Mathematics Series, Pitman Advanced Publishing Program, Boston, 1984.
[3] M. Sh. Birman, and V. V. Borzov, The asymptotic behavior of the discrete spectrum of certain singular differential operators, in “Problems of Mathematical
Physics, No. 5: Spectral Theory” (Russian), pp. 24–38. Izdat. Leningrad. Univ.,
Leningrad, 1971.; English transl. in: “Spectral Theory”, M. Sh. Birman (Ed.),
Topics in Mathematical Physics, Vol. 5., Consultants Bureau, New York, 1972.
[4] R. Carlson, Nonclassical Sturm-Liouville problems and Schrödinger operators on
radial trees, Electron. J. Differential Equations 2000 (2000), 1–24.
[5] W. D. Evans, and Y. Saito, Neumann Laplacians on domains and operators on
associated trees, Quart. J. Math. 51 (2000), 313–342.
[6] T. Kato, “Pertubation Theory for Linear Operators”, Springer-Verlag, Berlin,
1995.
[7] S. Kosugi, A semilinear elliptic equation in a thin network-shaped domain, J. Math.
Soc. Japan 52 (2000), 673–697.
[8] S. Kosugi, Semilinear elliptic equations on thin network-shaped domain with variable thickness, J. Differential Equations 183 (2002), 165–188.
[9] P. Kuchment, and H. Zeng, Asymptotics of spectra of Neumann Laplacians in
thin domains, in: “Advances in Differential Equations and Mathematical Physics”,
Yu. Karpeshina, G. Stolz, R. Weikard, and Y. Zeng (Eds.), Contemporary Mathematics AMS 387 (2003), 199–213.
[10] P. Kuchment, and H. Zeng, Convergence of spectra of mesoscopic systems collapsing onto a graph, J. Math. Anal. Appl. 258 (2001), 671–700.
[11] R. T. Lewis, Singular elliptic operators of second order with purely discrete spectra,
Trans. Amer. Math. Soc., 271 (1982), 653–666.
[12] B. B. Mandelbrot, “The Fractal Geometry of Nature”, W. H. Freeman and Co.,
New York, 1982.
[13] V. G. Maz’ja, “Sobolev Spaces”, Springer-Verlag, Berlin, 1985.
[14] K. Naimark, and M. Solomyak, Geometry of Sobolev spaces on regular trees and
the Hardy inequalities, Russ. J. Math. Phys. 8 (2001), 322–335.
[15] K. Naimark, and M. Solomyak, Eigenvalue estimates for the weighted Laplacian
on metric trees, Proc. London Math. Soc. 3 (2000), 690–724.
[16] T. R. Nelson, B. J. West, and A. L. Goldberger, The fractal lung: Universal and
species-related scaling patterns, Experientia 46 (1990), 251–254.
34
YEHUDA PINCHOVER, GERSHON WOLANSKY, AND DAPHNE ZELIG
[17] T. R. Nelson, and D. K. Manchester, Modeling of lung morphogenesis using fractal
geometries, IEEE Trans. Medical Imaging 7 (1988), 321–327.
[18] C. Puente-Baliarda, J. Romeu, R. Pous, and A. Cardama, On the behavior of
the Sierpinski multiband fractal antenna, IEEE Trans. Antennas Propagation 46
(1998), 517–524.
[19] C. Puente, J. Claret, F. Sagués, J. Romeu, M. Q. López-Salvans, and R. Pous,
Multiband properties of a fractal tree antenna generated by electrochemical deposition, IEE Electronics Lett. 32, (1996), 2298–2299.
[20] M. Reed, and B. Simon, “Methods of Modern Mathematical Physics”, vol. IV,
Academic Press, New York, 1980.
[21] J. Rubinstein, and M. Schatzman, Variational problems on multiply connected
thin strip I: Basic estimates and convergence of the Laplacian spectrum, Arch.
Rational Mech. Anal. 160 (2001), 271–308.
[22] M. Solomyak, Laplace and Schrödinger operators on regular metric trees: the
discrete spectrum case, in: “Function Spaces, Differential Operators and Nonlinear
Analysis, The Hans Triebel Anniversary Volume”, pp. 161–181, Birkhäuser, Basel,
2003.
[23] M. Solomyak, On the eigenvalue estimates for the weighted Laplacian on metric
graphs, in: “Nonlinear Problems in Mathematical Physics and Related Topics I,
In Honor of Professor O. A. Ladyzhenskaya”, M. Sh. Birman, S. Hildebrandt,
V. A. Solonnikov, and N. H. Uraltseva (Eds.), pp. 327–347, Kluwer, New York,
2002.
[24] E. R. Weibel, Design of airways and blood vessels considered as branching tree,
Chapter 74, in: “The Lung”, R. G. Crystal, J. B. West, and P. J. Barnes (Eds.),
Lippencott-Raven Inc., Philadelphia, 1997.
[25] D. Zelig, Properties of solutions of partial differential equations defined on human
lung-shaped domains, Ph. D thesis, Technion - Israel Institute of Technology, Israel,
2005.
Department of Mathematics, Technion - Israel Institute of Technology, Haifa
32000, Israel
E-mail address: [email protected]
Department of Mathematics, Technion - Israel Institute of Technology, Haifa
32000, Israel
E-mail address: [email protected]
Department of Mathematics, Technion - Israel Institute of Technology, Haifa
32000, Israel
E-mail address: [email protected]